confused about A080595

[Joshua Zucker]

Maybe I don't know what "lexicographic order" means correctly,
but I think that A080595 ought to be the third column of

0 0 0
0 1 1
0 2 2
0 3 3
0 4 4
0 5 5
and so forth ... hence a pretty boring sequence.

But it's given as
1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8,
not 0,1,2,3,4,5.

I wonder if maybe I'm supposed to sort the losing positions by sum of
the three numbers, and use the lexicographic ordering to break ties?

Then I think it should go
0 0 0
0 1 1
0 2 2
0 3 3
1 2 3
0 4 4
0 5 5
1 4 5

Ah ha!  (Well, except for the leading 0.)
So my conjecture is that this is the right sequence,
and that the name should be edited to say "sorted by sum of the three
numbers, and then lexicographically to break ties" or something like
that.

If that conjecture is correct, then the sequence goes
0 1 2 3 3 4 5 5 6 6 7 7 7 7 6 8 9 9 10 10 11 11 11 11 10 12 12 13 13
13 13 12 14 14 14 14 12 15 15 15 15 14 15 15 14 15 13 15 14 13 12 16
17 17 18 18 19 19 19 19 18 20 20 21 21 21 21 20 22 22 22 22 20 23 23
23 23 22 23 23 22 23 21 23 22 21 20 24 24 25 25 25 25 24 26 26 26 26
24 27 27 27 27 26 27 27 26 27 25 27 26 25 24 28 28 28 28 24 29 29 29
29 28 29 29 28 29 25 29 28 25 24.

BUT the given sequence has an 8 after the four 7s.

AND if my conjecture is correct, then the terms of A080594 should be
0 1 2 3 2 4 5 4 6 4 7 6 5 4 5 8 9 8 10 8 11 10 9 8 9 12 8 13 12 9 8 9
14 12 10 8 10 15 14 13 12 13 11 10 11 9 11 8 9 10 11 16 17 16 18 16 19
18 17 16 17 20 16 21 20 17 16 17 22 20 18 16 18 23 22 21 20 21 19 18
19 17 19 16 17 18 19 24 16 25 24 17 16 17 26 24 18 16 18 27 26 25 24
25 19 18 19 17 19 16 17 18 19 28 24 20 16 20 29 28 25 24 25 21 20 21
17 21 16 17 20 21
which they are not.

And similarly I would get A080593 to be
0 0 0 0 1 0 0 1 0 2 0 1 2 3 3 0 0 1 0 2 0 1 2 3 3 0 4 0 1 4 5 5 0 2 4
6 6 0 1 2 3 3 4 5 5 6 6 7 7 7 7 0 0 1 0 2 0 1 2 3 3 0 4 0 1 4 5 5 0 2
4 6 6 0 1 2 3 3 4 5 5 6 6 7 7 7 7 0 8 0 1 8 9 9 0 2 8 10 10 0 1 2 3 3
8 9 9 10 10 11 11 11 11 0 4 8 12 12 0 1 4 5 5 8 9 9 12 12 13 13 13 13

The first few triples, in my ordering, would be:
0 0 0
0 1 1
0 2 2
0 3 3
1 2 3
0 4 4
0 5 5
1 4 5
0 6 6
2 4 6
0 7 7
1 6 7
2 5 7
3 4 7
3 5 6
0 8 8
0 9 9
1 8 9
0 10 10
2 8 10
0 11 11
1 10 11
2 9 11
3 8 11
3 9 10

Can someone help me figure out what order I'm supposed to put them in?
Or if there's something else I'm doing wrong?

Thanks,
--Joshua Zucker