"A Puzzle A Day" - A Three Horse Race

This puzzle comes from the Puzzle A Day pad (it's meant to last a year; at this rate I'll be completing it in 2025).

The question is simple (but with a slight twist):  "How many ways can a three horse race finish, including ties?"

The simple answer (excluding ties) can be found by looking at all the possible combinations; if we call the horses A, B and C, then we have:

ABC
ACB
BAC
BCA
CAB
CBA

Six ways.

Now, refreshing the list and including the ties (which will be shown in brackets):

ABC
A(BC)
(AB)C
ACB
A(CB)
(AC)B
BAC
B(AC)
(BA)C
BCA
B(CA)

(BC)A

CAB
(CA)B
C(AB)

CBA
(CB)A
C(BA)
(CAB)

13 in total - there are three variations for each of the previous combinations, so ABC now becomes ABC, (AB)C and A(BC) - but we must now deduplicate A(BC) and A (CB), and so on.  The red text above shows a duplicate of a combination which has already been seen above it in the list.

And there's one where all three tie together (ABC).  The need to deduplicate the ties makes this question more complex than it appears at first sight, and so (as is often the case) care is taken to fully understand the question.