[seqfan] 2-dimensional arrays
rda at lemma-one.com
Sat Sep 27 14:53:09 CEST 2014
The example of a 2-dimensional array in the help page on format:
is a symmetric array (NIM addition), so it doesn’t make clear how you number the diagonals.
Having looked at some examples, I think the antidiagonals are intended to be listed top-to-bottom
rather than left-to-right. Indeed, A27 seems to make this clear: the lattice points are enumerated like this:
1 2 4 7 ...
3 5 8 ...
6 9 ...
rather than like this
1 3 6 10 ...
2 5 9 ...
4 8 ...
Is this the intention? If so, then Maybe the help page could usefully refer to A27.
To my shame, I have contributed three 2-d arrays
and haven’t been consistent: I used left-to-right in two of them
and top-to-bottom in the other one. A114327 is another example
where the left-to-right convention is implied by the formula given in the name.
See below for more examples where the left-to-right convention is used.
A related issue is this: many 2-d arrays enumerate features of some infinite list
of finite combinatorial structures. My inclination would usually be to have a row
(rather than a column) for each structure, so the rows are finitely non-zero.
E.g., if I was drawing Pascal’s triangle as a square array, I would draw it
as a lower-triangular array. A188147 is an example that follows the
opposite convention. There are lots of examples where the tabf format has been used
because otherwise there would be lots of zeroes. In examples like A188147 and
one I am currently working on the growth rate of the rows (in my convention)
is quite fast and it makes sense to use the tabl format. Is there a general rule
on whether to make the rows or the columns finite. I tried to find examples
of this by searching for:
keyword:tabl keyword:nice ~name:triangle ~name:triangular seq:0,0
This was somewhat inconclusive: it produced A55791 and A185287 as examples
where the author’s drawing of the array follows my convention and makes the rows finite.
But that isn’t what you see when you click the table link, so it looks like the author was
following the left-to-right convention for enumerating the antidiagonals.
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