Identifier
Values
[1] => [1,0] => [1,1,0,0] => [[2],[]] => 0
[1,2] => [1,0,1,0] => [1,1,0,1,0,0] => [[3],[]] => 0
[2,1] => [1,1,0,0] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[1,2,3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [[4],[]] => 0
[1,3,2] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[2,1,3] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[2,3,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [[2,2,2],[]] => 2
[3,1,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [[3,3],[]] => 2
[3,2,1] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [[3,3],[]] => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [[5,3],[1]] => 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2],[]] => 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7],[]] => 0
[] => [] => [1,0] => [[1],[]] => 0
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Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.