[seqfan] Multiset defining partitions

franktaw at netscape.net franktaw at netscape.net
Thu Mar 17 14:28:46 CET 2011

```I have just traced back from some recent submissions to
https://oeis.org/A176723, which contains a definition of "multiset
defining partitions". This definition seems to me to be seriously

The first problem is that the author is misusing the word "multiset".
For a multiset, as for a set, the identity of the elements is
significant. (See, e.g., http://en.wikipedia.org/wiki/Multiset ) Thus
{1,1,2} is a different multiset than {1,2,2} (although not different
from {1,2,1}).

Given this understanding, it is not the case that only some partitions
define multisets; the partitions are precisely the (finite) multisets
of positive integers. (We'll leave infinite multisets out of this
discussion.)

What Wolfdieter is calling "multisets" are apparently equivalence
classes of multisets, where one multiset can be turned into another by
"renaming" the elements. It might be nice if there was a simple term
for such an equivalence class, but I don't know of one. However, there
is a simple unique representation for each such equivalence class as
... a partition. Two such multisets are equivalent if they have the
same collection of frequencies; that collection is a multiset of
positive integers, and hence a partition.

This partition of the frequencies of a multiset I call its "signature";
see A115621.

What Wolfdieter has done in A176723 is to single out a representative
of each equivalence class, the first one that occurs in the
enumeration. (This does not depend on A-St vs Mma ordering of the
parititons; all other representatives of the class have a larger sum.)
Again, I don't know of any name for this property; for the moment I'll
call them "monotonic" partitions.

There have been a number of sequences added or awaiting approval which
represent properties of these equivalence classes of multisets, as
ordered by the monotonic partitions in A-St order. I would instead have
ordered them by their signatures (in A-St order); but I won't insist on
that.

I have some more minor complaints about A176723:

1) The first line of the examples is obscure. It looks like '|' is
being used to separate partitions with different numbers of parts,
while ',' separates those with the same number of parts. It would be
better to use ';' instead of '|'; and maybe include some explanation.
2) There should be an initial term 1 with offset 0, representing the
empty partition. Likewise, the condition n>=1 in the comment starting
"Partitions of n are written ..." should be omitted. This means that
other sequences based on these values should include n=0.
3) There is no need to mention triangular numbers in the definition of
monotonic partitions; it suffices to say that M is the largest part
size - or even omit it, and just note that the e[j] are monotonically
decreasing. The facts about the relation to triangular numbers should
be stated, but not as part of the definition.