Avoiding multiples of $p$ Let $p$ be a prime number and $P=\{1,2,...,p-1\}$   
In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$
only when we sum the last summand?   
For example let $p=7$ .
Clearly, $1+2+3+4+5+6$ is such a sum   (In fact there exist $408$ such sums)
  but $2+3+5+4+1+6$ is not, because already $2+3+5+4=2\cdot7$.  
We can see after a little investigation that if the total number of sums is $f(p)$,then   
$\frac{(p-1)!}{p-2}\leq f(p)\leq (p-1)!$ (equality holds iff $p=3$)
Is it possible to improve this (to find asymptotic bounds or -even better- something precise)?
Thanks in advance!
EDIT : It is possible to prove in an elementary way that for every $n\in \mathbb{N}$,  $\phi(n)\mid f(n)$
($\phi(n)$ is Euler's function)
 A: As I mentioned in the comments above, the number of permutations of elements $1,2,\dots,p$ (i.e., including $p$) is just by factor of $p-2$ larger than the amount in question (in fact, this is true for any odd $p$). Such permutations are now counted in http://oeis.org/A232663
Here is an explicit formula for the number of such permutations for a prime $p$, which I got by playing with inclusion-exclusion:
$$\sum_{A\in M_p} (-1)^{m+n}\cdot \frac{1}{n!} \cdot p^{n-\mathrm{rank}(A)} \cdot \prod_{j=1}^n (c_j-1)! \cdot \prod_{i=1}^m \binom{r_i}{a_{i1}, \dots, a_{in}},$$
where:


*

*$M_p$ is the set of matrices with nonnegative integer entries that sum up to $p$, with no zero rows or columns. Size of $M_p$ is given by http://oeis.org/A120733

*$A=(a_{ij})$ is a matrix of size $m\times n$ (i.e., $m$ and $n$ are respectively the number of rows and columns in $A$);

*$c_j = \sum_{i=1}^m a_{ij}$ is the sum of $j$-th column of matrix $A$;

*$r_i = \sum_{j=1}^n a_{ij}$ is the sum of $i$-th row of matrix $A$;

*matrix rank is computed over $\mathrm{GF}(p)$.
The formula may not be so useful due to its complexity but it's still a nice one. But the same reason I do not post here its derivation as it would take at least a couple of pages.
The formula has been numerically tested for $p=5$ and $p=7$.
