Identifier
Values
[2] => [1,1] => [1,1,0,0] => [2,1] => 0
[1,1] => [2] => [1,0,1,0] => [1,2] => 2
[3] => [1,1,1] => [1,1,0,1,0,0] => [2,3,1] => 0
[2,1] => [2,1] => [1,0,1,1,0,0] => [1,3,2] => 1
[1,1,1] => [3] => [1,0,1,0,1,0] => [1,2,3] => 3
[4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 1
[2,2] => [2,2] => [1,1,1,0,0,0] => [3,2,1] => 0
[2,1,1] => [3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 2
[1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 4
[5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1
[3,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[2,2,1] => [3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
[2,1,1,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 3
[1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 5
[6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 0
[5,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 1
[4,2] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0
[4,1,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 2
[3,3] => [2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 1
[3,1,1,1] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 3
[2,2,2] => [3,3] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 0
[2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2
[2,1,1,1,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 4
[1,1,1,1,1,1] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 6
[6,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => 1
[5,2] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => 0
[5,1,1] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => 2
[4,3] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[4,2,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 1
[4,1,1,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => 3
[3,3,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 1
[3,2,2] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0
[3,2,1,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2
[3,1,1,1,1] => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => 4
[2,2,2,1] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 1
[2,2,1,1,1] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 3
[2,1,1,1,1,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 5
[1,1,1,1,1,1,1] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 7
[5,3] => [2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => 0
[5,2,1] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => 1
[4,4] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 0
[4,3,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 1
[4,2,2] => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => 0
[4,2,1,1] => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => 2
[3,3,2] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 0
[3,3,1,1] => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
[3,2,2,1] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 1
[3,2,1,1,1] => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => 3
[2,2,2,2] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0
[2,2,2,1,1] => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 2
[2,2,1,1,1,1] => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => 4
[5,4] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => 0
[4,4,1] => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 1
[4,3,2] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => 0
[4,3,1,1] => [4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => 2
[4,2,2,1] => [4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,3,6,7,2] => 1
[3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[3,3,2,1] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 1
[3,3,1,1,1] => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 3
[3,2,2,2] => [4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => 0
[3,2,2,1,1] => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,4,7,3] => 2
[2,2,2,2,1] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 1
[2,2,2,1,1,1] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,5,4] => 3
[5,5] => [2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => 0
[4,4,2] => [3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => 0
[4,4,1,1] => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,5,4,3] => 2
[4,3,3] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => 0
[4,3,2,1] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,5,3,7,2] => 1
[3,3,3,1] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 1
[3,3,2,2] => [4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,5,2,1] => 0
[3,3,2,1,1] => [5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,6,4,3] => 2
[3,2,2,2,1] => [5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,3,7,2] => 1
[2,2,2,2,2] => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => 0
[2,2,2,2,1,1] => [6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,4,3] => 2
[4,4,3] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,4,6,3,2,1] => 0
[3,3,3,2] => [4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,5,4,2,1] => 0
[3,3,3,1,1] => [5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 2
[3,3,2,2,1] => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,6,3,2] => 1
[2,2,2,2,2,1] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,3,2] => 1
[4,4,4] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 0
[3,3,3,3] => [4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,4,3,2,1] => 0
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Description
The aix statistic of a permutation.
According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then
$$ \operatorname{aix}\pi = \begin{cases} \operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\ 1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\ 0 & \text{ if } \beta = \emptyset \end{cases}\ . $$
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.