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Matching statistic: St000704
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Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000406
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 93%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 93%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1,2,2,2],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,1,2,2,2,2],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,1,1,2,2,3],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,1,2,2,2,3],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,1,2,2,3,3],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,2,2,2,2,3],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,2,2,2,3,3],[3,3],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
[[1,1,1,1,2,2,2],[3,4],[4]]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? = 2 - 1
Description
The number of occurrences of the pattern 3241 in a permutation.
Matching statistic: St000045
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000045: Binary trees ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000045: Binary trees ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,1,1,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,1,1,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,1,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,1,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,1,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,2,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,2,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,2,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[[1,1,1,1,1,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[[1,1,1,1,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[[1,1,1,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ? = 8
Description
The number of linear extensions of a binary tree.
Also, the number of increasing / decreasing binary trees labelled by $1, \dots, n$ of this shape.
Also, the size of the sylvester class corresponding to this tree when the Tamari order is seen as a quotient poset of the right weak order on permutations.
Also, the number of permutations which give this tree shape when inserted in a binary search tree.
Also, the number of permutations which increasing / decreasing tree is of this shape.
Matching statistic: St001289
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001289: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001289: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,1,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8
Description
The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero.
This n-fold tensor product seems to be always injective.
Matching statistic: St000217
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000217: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000217: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,1,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000218
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,1,1,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,1,1,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,1,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,1,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,2,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,2,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,2,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 1 - 1
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[[1,1,1,1,1,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[[1,1,1,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => ? = 8 - 1
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000426
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000426: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000426: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,2,2,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,2,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,2,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,1,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 8 - 1
Description
The number of occurrences of the pattern 132 or of the pattern 312 in a permutation.
Matching statistic: St001685
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1],[3],[4]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[3],[4]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1],[3],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1],[4],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[3],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[4],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[3],[4],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1 - 1
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1 - 1
[[1,1,1,1,1],[2,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,1],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[2,3]]
=> [5,7,1,2,3,4,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[2,2]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[2,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[2,3]]
=> [4,7,1,2,3,5,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[2,3]]
=> [4,6,1,2,3,5,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[2,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,2],[2,3]]
=> [3,7,1,2,4,5,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,3],[2,3]]
=> [3,6,1,2,4,5,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,3,3],[2,3]]
=> [3,5,1,2,4,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,3,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,3,3,3],[2,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,2],[2,3]]
=> [2,7,1,3,4,5,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,3],[2,3]]
=> [2,6,1,3,4,5,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,3,3],[2,3]]
=> [2,5,1,3,4,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,3,3,3],[2,3]]
=> [2,4,1,3,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [7,6,1,2,3,4,5] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[2],[3]]
=> [7,5,1,2,3,4,6] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[2],[3]]
=> [6,5,1,2,3,4,7] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[2],[3]]
=> [7,4,1,2,3,5,6] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[2],[3]]
=> [6,4,1,2,3,5,7] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,4,1,2,3,6,7] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St001705
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001705: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001705: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1],[3],[4]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[3],[4]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1],[3],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1],[4],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[3],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[2],[4],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[3],[4],[5]]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0 = 1 - 1
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1 - 1
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 1 - 1
[[1,1,1,1,1],[2,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,1],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[2,3]]
=> [5,7,1,2,3,4,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[2,2]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[2,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[2,3]]
=> [4,7,1,2,3,5,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[2,3]]
=> [4,6,1,2,3,5,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[2,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,2],[2,3]]
=> [3,7,1,2,4,5,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,3],[2,3]]
=> [3,6,1,2,4,5,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,3,3],[2,3]]
=> [3,5,1,2,4,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,3,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,3,3,3],[2,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,2],[2,3]]
=> [2,7,1,3,4,5,6] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,3],[2,3]]
=> [2,6,1,3,4,5,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,3,3],[2,3]]
=> [2,5,1,3,4,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,3,3,3],[2,3]]
=> [2,4,1,3,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[2,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [7,6,1,2,3,4,5] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,1,2],[2],[3]]
=> [7,5,1,2,3,4,6] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,1,3],[2],[3]]
=> [6,5,1,2,3,4,7] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,2,2],[2],[3]]
=> [7,4,1,2,3,5,6] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,2,3],[2],[3]]
=> [6,4,1,2,3,5,7] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,4,1,2,3,6,7] => [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
Description
The number of occurrences of the pattern 2413 in a permutation.
Matching statistic: St001314
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 79%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 79%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1,1,1],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,3,3,3],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[2,2]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[2,2]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[2,2]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[2,2]]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,4],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,4],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,4,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,4,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,4],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,4,4],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,3],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,4],[3],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,4,4],[2],[3]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,2,4,4],[2],[4]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
The following 124 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001960The number of descents of a permutation minus one if its first entry is not one. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001597The Frobenius rank of a skew partition. St001330The hat guessing number of a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000068The number of minimal elements in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001866The nesting alignments of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000907The number of maximal antichains of minimal length in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St000717The number of ordinal summands of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001490The number of connected components of a skew partition. St000910The number of maximal chains of minimal length in a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000307The number of rowmotion orbits of a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001754The number of tolerances of a finite lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000635The number of strictly order preserving maps of a poset into itself. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001625The Möbius invariant of a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000080The rank of the poset. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001645The pebbling number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000189The number of elements in the poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000400The path length of an ordered tree. St000180The number of chains of a poset. St000529The number of permutations whose descent word is the given binary word. St000100The number of linear extensions of a poset. St000416The number of inequivalent increasing trees of an ordered tree. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001909The number of interval-closed sets of a poset. St000410The tree factorial of an ordered tree. St000634The number of endomorphisms of a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St000422The energy of a graph, if it is integral.
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