Are there enough additive permutations? I am hoping to learn enough about additive permutations to help with a number theory problem.  These permutations also have connections to difference sets, orthomorphisms, transversals, and other structures.  As a meta question, I would like to know more references or applications for additive permutations, especially applications involving combinatorial number theory.  First, the basic setup.
Let $l$ be a nonegative integer, and let $k$ stand for both the set $\{t : t$ is an integer and $ \mid t \mid \leq l \}$ and the cardinality of the set $k=2l+1$.  I will represent the set of permutations $S_k$
 on the set $k$ by vectors indexed by $k$ in increasing order.  So for $l=2$, the
identity permutation $e$ is written as $\langle -2, -1 ,0 , 1, 2 \rangle$.  Let me take
$\pi \in S_k$ and write it as a vector, and I will write vector addition as $++$. I can then write $e  ++ \pi$ as a vector in $\mathbb{R}^k$ with $i$th coordinate being $i + \pi(i)$.  This vector $e ++ \pi$ is a vector with integer coordinates, and may not look anything like a vector representation of an element of $S_k$.  However, sometimes it does, and when this happens, we call $\pi$ an additive permutation.
As an example when $l=1$, one has two additive permutations $\langle 1,-1, 0 \rangle$ and
$\langle 0 , 1, -1 \rangle$, each of which is the negative reverse of the other ($\pi(i) = -\rho(-i)$ for all $i \in k$).  Except when $l=0$, the identity permutation $e$ is not an additive permutation.  Also, the fact that $\langle 1,0, -1 \rangle$ is not an additive permutation shows that this definition depends on representation: it cannot be defined as a characteristic subset of the permutation group on $k$ members.
The OEIS entry A002047 contains some references to the literature, which I am slowly absorbing.  However, I don't see the answer to either of the following questions:


*

*1) Given $l$, how many members of $S_k$ are additive permutations?  I have not found
an asymptotic formula, although the paper by Cavenagh and Wanless suggests an
exponential lower bound.  I have a weak upper bound which for most $l$ is slightly better than $l^{2l}$.

*2) Using just the group operation of the symmetric group $S_k$ (so no inverse, but $k$ is finite so inverse is not needed), do the additive permutations generate $S_k$?  It seems to be so for $l=0,1,2$.  (It is less interesting to me but still valid to ask for $l$ large if $S_k$ is generated using $++$.)
In addition to the OEIS references, I am perusing work of D.G. Rogers, and am open to other suggestions for references.  I am also looking at a related paper, but the operation $++$ there is over a finite ring, and I don't think I can use those results yet.
Gerhard "First Question On This Account" Paseman, 2015.07.10
 A: I want to share a partial answer to question 1), and raise a few more questions.  I found what I think is a neat and likely unoriginal bijection; I'm hoping the combinatorialists can provide a reference and perhaps use it to help with my questions above.
I decided to try a $k$ by $k$ chessboard visualization of the enumeration problem, and succeeded, sort of.  I had to cut off triangular pieces of the chessboard and ask for a maximal nonattacking and covering queen placement, except that the queens were restricted in movement relative to actual queens in chess.  So I shifted that set up to a hexagonal board, and invented a (name for a likely unoriginal) piece called a "jack", which is like what a rook would be for a board of hexagonal cells (from a hexagonal cell, move in one of the 6 directions perpendicular to a side of the cell, along 3 lines which I call "diagonals"), but decided it needed to be partly royal.
Let's imagine a hexagonal array with $l+1$ cells on a side, with vertex cells at the even numbers on a clock, and for orientation label the six vertex cells A,B,C,D,E, and F in clockwise fashion.  (So A is at 12, B is at 2, and D is at 6 o'clock.)  Now, starting with the vertex cell E and going clockwise up to A, label (just outside the board) those cells with $-l, -l+1, \ldots$ all the way up to $l$, so that the $k$ "diagonals" from upper left to lower right are labeled with the $\pi(i)$ index. Cell F will have a $0$ label for $\pi(0)$ and cell A will have a $l$ label for $\pi(l)$.
In a similar fashion, we label the other diagonals from $-l$ at Cell A clockwise through $0$ for Cell B and ending with $l$ at cell C.  We have labeled these with $j$- values, which will represent values being placed in position $\pi(i)$.
If we look at a cell, it belongs to exactly two of the labeled diagonals, $\pi(i)$ and $j$
say.  If we were to decorate all the cells with the value $i+j$ when it is on diagonals
$\pi(i)$ and $j$, we would see constant values running up and down.  In particular, the
line of cells from A down to D would get the value $0$, $-l$ for the cells between (and including) cell E and F, and $l$ for cells B to C.
Once we have the observation of the values being constant on the "third" (vertical) diagonal, I can now assert the bijection.  Let G be a placement of $k$ jacks on this board
that do not attack each other and consequently cover all cells of the board.  Since I have
labeled the diagonals, I will call such a G a labeled placement.
Assertion:  The number of labeled placements on this $k$ board is the same as the number of additive permutations of $S_k$.
For an additive permutation $\rho \in S_k$ place a jack at cell on diagonals $\pi(\rho(j))$ and $j$ for each $j \in k$.  After I have verified the details, I will call the Assertion a Proposition.  However, I don't see what could go wrong, yet.
I am willing to shorten this description if someone will provide a graphic version.  Now for the payoff:  Referring to the third diagonal, place a jack on the 0 diagonal somewhere in one of the $k$ cells; this will leave usually $k-3$ and at most $k-2$ uncovered cells on the 1 diagonal to place another jack, so place one there.  We get an upper bound of $k!!$ placements of those $l+1$ mutually nonattacking jacks, and $l!$ trivially for the remaining $l$ jacks, giving an upper bound of $k!/2^l$ for the number of additive permutations in $S_k$.
I am still working on the lower bound, but have a feeling that $l!$ or even $(l+1)!$ might be achievable with the Assertion and this picture.  Now for the additional questions:


*

*3) has anyone seen this bijection before, and will they please give me a reference?

*4)  has labeled placement of nonattacking jacks on a hexagonal board appeared before, hopefully with enumeration?

*5) assuming I did not screw up and the Assertion is soon to be a Proposition, can anyone
see a good lower bound (better than exponential, and hopefully factorial) for the number
of additive permutations with this or any other picture?

*6) leaving additive permutations aside, it is tempting to view the board as a representation of a three-dimensional cubical array.  Is there a combinatorial advantage to such a perspective for this problem?  That is, picking $k$ cubes out of an array of $l+1$ cubes on a side, does this count or represent a nice entity in finite geometry or some other field?
I suspect that a factorial lower bound would imply improvements on the current literature.
I also would not mind improvements on the upper bound.
Gerhard "Likes Looking At Suggestive Pictures" Paseman, 2015.07.12
