Data structure for ordered counted set Is there a name for a counted set (multiset) that is ordered?

For example lets say this data structure represents a shopping cart (or basket if you're British). The shopping cart shows the order the items were added to the cart unless an item of the same type was already added in which case a number associated with item is incremented.

Is this a well studied data structure or just a specialized multiset? I imagine it could be represented with an internal data structure of a dictionary and an array.
Just use a regular dictionary (also dynamic set) and attach a counter to each entry. The insert operation then finds already existing entries and can increase the counter; similarly for delete.

If you want to search the literature, some sources (e.g. Algorithms by R. Sedgewick and K. Wayne (2011, 4th ed)) call the abstract data structure for multi sets a bag. This notion seems to be used in practice as well. 

Note that some implementations will assume that you assign data to keys, and that the same key can occur multiple times with different data. In this case, using a counter is not enough, so these may not directly do what you want (since they may not even bother to detect duplicate keys).