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

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