Identifier
Values
[1] => [1,0] => [1,1,0,0] => [2,1] => 1
[2] => [1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => 3
[1,1] => [1,1,0,0] => [1,1,1,0,0,0] => [3,2,1] => 2
[3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 6
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => 4
[1,1,1] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => 3
[4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 10
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 7
[2,2] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 5
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 5
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 15
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => 11
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 8
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => 8
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 6
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => 6
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => 5
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => 12
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 9
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => 9
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 7
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => 7
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,6,4,5,3,1] => 13
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => 10
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => 10
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => 8
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => 14
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => 11
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => 9
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 12
[] => [] => [1,0] => [1] => 0
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Description
The Denert index of a permutation.
It is defined as
$$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is Euler-Mahonian. Here, $exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
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.