How many perfect secrecy systems are there? How many non-trivial*, interesting perfect secrecy systems are there other than the one-time-pad?  Does it seem that the one-time-pad and perfect secrecy are synonymous, but are there any other cryptosystems that have perfect secrecy aside from the trivial examples?

*Here the non-trivial means; the simple type that may be created by, say, picking a set of integers, and randomly corresponding each of those integers with a 0 or 1 (the key), then sending some permutation of the integers as cipher-text.
How many non-trivial*, interesting perfect secrecy systems are there other than the one-time-pad? 


Infinitely many.

Let $\mathbb G$ be a group (written multiplicatively). Then


$\operatorname{KeyGen}(1^n)=k\stackrel{\$}{\gets}\mathbb G^n$, that is the key is a vector of $n$ independently random group elements.
$\operatorname{Enc}(k,m):\mathbb G^n\times \mathbb G^n\to\mathbb G^n:(k,m)\mapsto m\cdot k$, that is the plaintext is a vector of group elements and the ciphertext is simply the element-wise multiplication of each message element with the corresponding key element.
$\operatorname{Dec}(k,m):\mathbb G^n\times \mathbb G^n\to\mathbb G^n:(k,c)\mapsto c\cdot k^{-1}$, that is we simply combine each ciphertext element with the inverse of the corresponding key element.


Note that the above defines an encryption scheme for each group there is and that's at least countably infinitely many ones (as $(\mathbb Z_n,+)$ works for all $n\in\mathbb N$). In particular it also works for non-abelian (i.e. non-commutative) groups.

The above scheme can be proven secure under a similar argument to the traditional one-time-pad and in fact the traditional one is simply this with the group $(\mathbb Z_2,+)$.