[seqfan] Qualifying the counting of triangles

[Peter Munn]

In sequences which count triangles, is it best to clarify that mirror
images are not counted separately and, if so, how?

This arises because when adding a CROSSREF to A054875, I thought its name
"Number of congruent triangles with integer sides and [other conditions]"
was unclear, so asked if it 'could be clarified, perhaps the same
"congruent" -> "pairwise incongruent" amendment that its companion
sequence, A054876, had in 2012'.

With editorial encouragement, I made the change; but my later OEIS
sequence search for "pairwise incongruent" found it occurred elsewhere
only in the phrase "pairwise incongruent modulo ...".

So when my contribution to A054875 lead to my co-authoring its derived
records sequence, A322105, I made a brief almost fruitless search of other
triangle counting sequences to find suitable wording. First, I decided on
this COMMENT: "Congruent triangles are identified", then an editorial pink
box response lead me to step back and research further...

I find sequences that count other geometric entities use various
words/phrases to clarify whether mirror images (and sometimes rotations)
are counted separately: fixed; free; one-sided; distinguishing/identifying
mirror images; etc. [1].

Triangle-counting sequences generally identify mirror images implicitly.
This might partly be because many equivalent sequences that distinguish
mirror images are not in OEIS. For instance, I haven't found the
equivalents for A054875 ("2,1,0,1,0,2,0,1,3"?) and A051493
("1,0,1,0,2,1,3,1,5,3,7,3"?).

Especially as such sequences would presumably qualify for inclusion in
OEIS, what would you think is best practice for indicating how mirror
images are handled?

Best Regards,

Peter


[1] The qualifications "fixed" and "free" are the most common, usually
used where specific rotations are possible too. The phrases
"distinguishing/identifying mirror images" occur in George Sicherman's
contributions on counting polyaboloes, polydrafters, polyominoes,
polycubes and polytets.

The phrase "one sided" appears quite often, including for: polyominoes (at
least 4 sequences); n-iamonds, polydrafters, polyhexes and polytiles (7);
similar entities using non-planar geometries (4); arrangements of circles
(2); and prudent walks (6).  In constrast, "two-sided" appears in a name,
only in sequences counting prudent walks.

Other variations include: the first comment of A008742 which speaks of
"triangles [...] modulo rotations but not reflections";  A288568 and
A296406-12 (arrangements of pseudo-circles on a sphere) use the words
"reduced for mirror symmetry";  A060312 says "solutions are identified if
they are rotations or reflections of each other".

I can provide further details of some of the above on request.