[seqfan] Re: Please help to check A074140.

[Peter Luschny]

EJ> since ordering is usually taken to be irrelevant.

Right. But for some sequences ordering is of course relevant.
And the results may be different if you use a CAS like Maple
or Mathematica and just call the partition generating functions
provided. They might be implemented with different orderings
and indeed they are.

EJ> Why such fancy and confusing terminology is necessary
for all of this is beyond me.

I once tried to approach it in a systematic way by relating
all the orderings to one special tree, the partition tree as
generated by the Fenner-Loizou algorithm (in part because it
is described by Knuth in TAOCP 4). Then you have immediately
6 orderings with regard to the traverse, two ways for duality
(conjugates), two for monotony conditions (increase/decrease),
two for forward/reverse listing, thus about 50 ways to present
the partitions in a linear order. So there is a certain need for a
consistent terminology. But the terminology provided in the
Wiki (and in the comments in OEIS) looks totally unusable
for this purpose.

Peter

http://oeis.org/wiki/User:Peter_Luschny/IntegerPartitionTrees