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Your data matches 87 different statistics following compositions of up to 3 maps.
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Matching statistic: St000175
St000175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 2
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 0
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 6
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 3
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000766
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [2] => 0
[1,1]
=> [[1],[2]]
=> [1,1] => 0
[3]
=> [[1,2,3]]
=> [3] => 0
[2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[4]
=> [[1,2,3,4]]
=> [4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [5] => 0
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [6] => 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 0
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 5
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 6
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => 0
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => 3
Description
The number of inversions of an integer composition.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
Matching statistic: St000769
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [2] => [2] => 0
[1,1]
=> [[1],[2]]
=> [1,1] => [1,1] => 0
[3]
=> [[1,2,3]]
=> [3] => [3] => 0
[2,1]
=> [[1,2],[3]]
=> [2,1] => [2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 0
[4]
=> [[1,2,3,4]]
=> [4] => [4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [3,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2,2] => 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [5] => [5] => 0
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [4,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [3,2] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,3,1] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,2,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [6] => 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [5,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [4,2] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,4,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [3,2,1] => 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,1,3,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [2,2,2] => 0
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,1,2,1] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,1,1,2,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [7] => 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [6,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [5,2] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,5,1] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [4,3] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [4,2,1] => 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,1,4,1] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [3,3,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [2,3,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,3,2,1] => 5
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,1,1,3,1] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [2,2,2,1] => 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [1,2,1,2,1] => 6
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,1,1,1,2,1] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [8] => 0
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [7,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [6,2] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,6,1] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [5,3] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [5,2,1] => 3
Description
The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see [[St000008]].
Matching statistic: St000599
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000599: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000599: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Values
[1]
=> [1,0]
=> {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3,6},{4},{5}}
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2,6},{3},{4},{5}}
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,6},{2},{3},{4},{5}}
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6,7}}
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,7},{6}}
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,6},{4,5}}
=> 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,7},{5},{6}}
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2,6},{3,4},{5}}
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,7},{4},{5},{6}}
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1,6},{2,3},{4},{5}}
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,7},{3},{4},{5},{6}}
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,7}}
=> 8
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4,6},{5}}
=> 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4,7},{5,6}}
=> 7
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2,6},{3,5},{4}}
=> 3
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3,7},{4,5},{6}}
=> 5
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6,7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,9},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,9},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,9},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
Description
The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block.
Matching statistic: St000612
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Values
[1]
=> [1,0]
=> {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3,6},{4},{5}}
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2,6},{3},{4},{5}}
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,6},{2},{3},{4},{5}}
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6,7}}
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,7},{6}}
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,6},{4,5}}
=> 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,7},{5},{6}}
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2,6},{3,4},{5}}
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,7},{4},{5},{6}}
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1,6},{2,3},{4},{5}}
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,7},{3},{4},{5},{6}}
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,7}}
=> 8
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4,6},{5}}
=> 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4,7},{5,6}}
=> 7
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2,6},{3,5},{4}}
=> 3
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3,7},{4,5},{6}}
=> 5
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6,7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,9},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,9},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,9},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000491
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Values
[1]
=> [1,0]
=> {{1}}
=> {{1}}
=> ? = 0
[2]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[1,1]
=> [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> {{1,6},{2},{3},{4},{5}}
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> {{1},{2,6},{3},{4},{5}}
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,5},{3}}
=> 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3,6},{4},{5}}
=> {{1},{2},{3,6},{4},{5}}
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2,6},{3},{4},{5}}
=> {{1},{2},{3},{4,6},{5}}
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,6},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5,6}}
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6,7}}
=> {{1,7},{2},{3},{4},{5},{6}}
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> {{1,5,6},{2},{3},{4}}
=> 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,7},{6}}
=> {{1},{2,7},{3},{4},{5},{6}}
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,6},{4,5}}
=> {{1,5},{2,6},{3},{4}}
=> 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,7},{5},{6}}
=> {{1},{2},{3,7},{4},{5},{6}}
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> {{1},{2,4},{3,5}}
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2,6},{3,4},{5}}
=> {{1,4},{2},{3,6},{5}}
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2},{3},{4,7},{5},{6}}
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1,6},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5,6}}
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,7},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5,7},{6}}
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6,7}}
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,7}}
=> {{1,6,7},{2},{3},{4},{5}}
=> 8
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4,6},{5}}
=> {{1,4},{2,6},{3},{5}}
=> 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4,7},{5,6}}
=> {{1,6},{2,7},{3},{4},{5}}
=> 7
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2,6},{3,5},{4}}
=> {{1},{2,5},{3,6},{4}}
=> 3
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> {{1,4,5,6},{2},{3}}
=> 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3,7},{4,5},{6}}
=> {{1,5},{2},{3,7},{4},{6}}
=> 5
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? ∊ {0,0,2,2,3,4,5,6}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6,7}}
=> {{1,7},{2,8},{3},{4},{5},{6}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,9},{7},{8}}
=> {{1},{2},{3,9},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> {{1,6},{2},{3,8},{4},{5},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,9},{6},{7},{8}}
=> {{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4,5},{6},{7}}
=> {{1,5},{2},{3},{4,8},{6},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5,9},{6},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> {{1,4},{2},{3},{5},{6,8},{7}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,9},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6,9},{7},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> {{1,3},{2},{4},{5},{6},{7,8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? ∊ {0,0,1,1,3,3,4,5,5,5,6,7,7,9,10}
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000355
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 64%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 64%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,1,2,3,4,5}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => ? ∊ {0,1,2,3,4,5}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,3,5,6] => 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => ? ∊ {0,1,2,3,4,5}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => ? ∊ {0,1,2,3,4,5}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => ? ∊ {0,1,2,3,4,5}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,1,2,3,4,5}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [4,1,2,3,5,6,7,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => 3
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,1,5,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [6,1,2,3,4,5,7,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,6,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [3,2,1,4,5,6,7,8] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [4,2,1,3,5,6,7,8] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => 2
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1,5,6,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,1,4,6,7,8] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => 3
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,4,1,2,5,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,1,5,7,8] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,1,5,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [7,2,3,4,5,1,6,8] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,1,6] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,5,6,1,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000039
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 64%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 64%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0,1,2,3,4,5}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? ∊ {0,0,1,2,3,4,5}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? ∊ {0,0,1,2,3,4,5}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1,2,7,3,4,5] => ? ∊ {0,0,1,2,3,4,5}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,7,4,5] => ? ∊ {0,0,1,2,3,4,5}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => ? ∊ {0,0,1,2,3,4,5}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? ∊ {0,0,1,2,3,4,5}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,1,7,2,3,4] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [7,1,2,8,3,4,5,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => 3
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,1,2,7,3,5] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1,2,3,8,4,5,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,8,5,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,1,3,4,5,7] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => ? ∊ {0,1,1,2,2,3,3,4,5,5,6,7,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [6,7,1,8,2,3,4,5] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => 2
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,1,6,2,7,3,4] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [5,7,1,2,8,3,4,6] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => 3
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,1,6,2,3,7,5] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,1,7,2,3] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [4,7,1,2,3,8,5,6] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,1,6,2,4,5,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,7,5] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,7,1,2,4,5,8,6] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,1,4,5,7] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,7,1,3,4,5,6,8] => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,1,1,1,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000800
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 64%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 64%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0,2,3,3,4,5}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? ∊ {0,0,2,3,3,4,5}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? ∊ {0,0,2,3,3,4,5}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1,2,7,3,4,5] => ? ∊ {0,0,2,3,3,4,5}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1,2,3,7,4,5] => ? ∊ {0,0,2,3,3,4,5}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => ? ∊ {0,0,2,3,3,4,5}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? ∊ {0,0,2,3,3,4,5}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => 3
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,1,7,2,3,4] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [7,1,2,8,3,4,5,6] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => 3
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,1,2,7,3,5] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,1,2,3,8,4,5,6] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,8,5,6] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,1,3,4,5,7] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => ? ∊ {0,1,1,2,2,3,4,4,5,5,6,7,8}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [6,7,1,8,2,3,4,5] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,1,6,2,7,3,4] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [5,7,1,2,8,3,4,6] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,1,6,2,3,7,5] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,1,7,2,3] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [4,7,1,2,3,8,5,6] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 2
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,1,6,2,4,5,7] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,7,5] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,7,1,2,4,5,8,6] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,1,4,5,7] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,7,1,3,4,5,6,8] => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? ∊ {0,0,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
Description
The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000585
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000585: Set partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 64%
Mp00284: Standard tableaux —rows⟶ Set partitions
St000585: Set partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 64%
Values
[1]
=> [[1]]
=> {{1}}
=> ? = 0
[2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 3
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 4
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 2
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 5
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 3
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> 6
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> {{1,2,5,6,7,8},{3,4}}
=> 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,0,1,2,2,3,3,3,4,4,5,5,5,6,6,7,8}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> {{1,2,5,6,7,8,9},{3,4}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> {{1,4,5,6,7,8,9},{2},{3}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> {{1,2,3,7,8,9},{4,5,6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> {{1,3,6,7,8,9},{2,5},{4}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> {{1,5,6,7,8,9},{2},{3},{4}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> {{1,2,3,4,9},{5,6,7,8}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> {{1,3,4,8,9},{2,6,7},{5}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> {{1,2,7,8,9},{3,4},{5,6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> {{1,4,7,8,9},{2,6},{3},{5}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> {{1,6,7,8,9},{2},{3},{4},{5}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> {{1,3,4,5},{2,7,8,9},{6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> {{1,5,8,9},{2,7},{3},{4},{6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> {{1,5,6},{2,8,9},{3},{4},{7}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> {{1,2,9},{3,4},{5,6},{7,8}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> {{1,6,9},{2,8},{3},{4},{5},{7}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> {{1,7},{2,9},{3},{4},{5},{6},{8}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,8,9,9,10}
Description
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.
The following 77 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001781The interlacing number of a set partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000358The number of occurrences of the pattern 31-2. St000222The number of alignments in the permutation. St001535The number of cyclic alignments of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000356The number of occurrences of the pattern 13-2. St000456The monochromatic index of a connected graph. St000422The energy of a graph, if it is integral. St001867The number of alignments of type EN of a signed permutation. St001330The hat guessing number of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001846The number of elements which do not have a complement in the lattice. St000177The number of free tiles in the pattern. St001498The normalised height of a Nakayama algebra with magnitude 1. St001556The number of inversions of the third entry of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000516The number of stretching pairs of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001520The number of strict 3-descents. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001822The number of alignments of a signed permutation. St001964The interval resolution global dimension of a poset. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000527The width of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000488The number of cycles of a permutation of length at most 2. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001856The number of edges in the reduced word graph of a permutation. St001868The number of alignments of type NE of a signed permutation. St001948The number of augmented double ascents of a permutation. St000902 The minimal number of repetitions of an integer composition. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001530The depth of a Dyck path. St001866The nesting alignments of a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000068The number of minimal elements in a poset. St000264The girth of a graph, which is not a tree. St000534The number of 2-rises of a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000679The pruning number of an ordered tree.
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