[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 

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 

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.

Franklin T. Adams-Watters

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