[seqfan] Re: Please help to check A074140.

[franktaw at netscape.net]

At one time, we had simply the A&S ordering and the Mathematica 
ordering. Other orderings were defined in terms of these where 
possible. Objections were raised to both of these names, on the grounds 
that neither was the original source of that ordering. (They are, in  
fact, the immediate sources for these orderings in the OEIS.) The 
standard was that sequences presenting some property of each partition 
should be present in both orders. A start was made at trying to adhere 
to this standard, but it didn't get very far. (Neil asked me at the 
time to make the effort, but I did not have sufficient free time 
available to take it on.)

This was replaced with the current terminology by someone. In my mind, 
this was a step backwards: it introduces obscure descriptive words, 
which are neither intuitive nor used in the literature; and it 
conflates the order of the partitions with the order of presentation of 
the parts of each partition.

I am not in a position to do a literature search, but it does not 
appear that there are standard names in the literature for any of these 
orderings. If someone can find that there are, we should certainly 
adopt them (and add appropriate references). Otherwise, we should 
decide on names we consider appropriate for our two primary orderings, 
perhaps by finding the first publication of each, and use those.

Franklin T. Adams-Watters

-----Original Message-----
From: Peter Luschny <peter.luschny at gmail.com>

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

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