[seqfan] Re: Please help to check A074140.

[L. Edson Jeffery]

Peter,

For as close to "context-free" as one might get, see "An Introduction to
the Theory of Numbers" by Hardy and Wright, where an intuitive description
of the algorithm is given but not formalized (since ordering is usually
taken to be irrelevant). I have the fifth edition (1979) of the book, but
the algorithm (which appears on page 273) is probably described as far back
as the first edition from 1938 and likely did not originate even there.

Why such fancy and confusing terminology is necessary for all of this is
beyond me. However, it appears that "Abramowitz and Stegun" just trivially
reverses the order of the parts (not the partitions) of "Hardy and Wright"
if that makes sense.

Example partitions of 4:

Hardy and Wright:      {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}

Abramowitz and Stegun: {4}, {1,3}, {2,2}, {1,1,2}, {1,1,1,1}

The conjugates of "Hardy and Wright" just give rise to the list of
partitions (not the parts) taken in reverse order:

{1,1,1,1}, {2,1,1}, {2,2}, {3,1}, {4}

which can be realized from certain rectangular arrays of dots described
beginning on the same page (273) in the above book.

Ed Jeffery

>I have a hard time to understand this. I would like to have a
*context-free* definition of all this stuff, not a heap of terms pointing
to one another. The very position of the term 'reverse' in these
descriptions often makes several readings possible.

>It would be very helpful to see algorithms which describe the meaning of
all these terms in a precise, unique way.

>Peter