How to store variable length arrays in a database - php

Question

I'm looking for a way to save data based off a set of rows in another table, but I don't know how to set up the field. Think something similar to using the results of a mysql group_concat as the field. The data is based off the unique combination of rows, rather than one row or field.
What I need to be able to do is:
Store the array itself in the database
Store associated data about the array
Retreive the array
The ability to lookup data about the array using the data in the array
Some options I've thought about:
Saving as an ordered set concatenated into a string.
Saving the serialized array (serialized using php's serialize function).
Saving the set as a hashed string using a reversible hash.
None of these options seem correct so I came here hoping someone has a better answer.
Background:
Supposed I have the following tables:
users {id, other unimportant fields}
products {id, other unimportant fields}
shipments {id, user_id, product_id, date, other unimportant fields}
I want to create a new table called assigned_products where the assigned product is based off of the unique combination of products they've received in the past. So assigned_products should look like:
assigned_products {set_of_products_received (array), product_id (data about the array)}
I don't know of a good way to store set_of_products_received in a database.
Example use:
Suppose I have 100 users who got product A, 100 users who got product B, and 100 users who got products A and B. Suppose then I wanted to give product B to everyone who got product A, product A to everyone who got product B, and product C to everyone who got product A and B. The assigned products table should look like:
+--------------------------+------------+
| set_of_products_received | product_id |
+--------------------------+------------+
| A | B |
| B | A |
| A, B | C |
+--------------------------+------------+
I'm just looking for a better way of storing set_of_products_received
Reading this over I realize it's a bit hard to understand, but I don't really know the appropriate terms to describe this issue (probably why I'm having trouble finding solutions). I'll be happy to clarify if anyone has any questions.