"unions" of multisets?
Michael J Hardy
multiplicity is a positive integer. Or maybe non-negative;
you can do it either way if you're consistent. The simplest
natural example is the prime factorization of a natural
number. My first question is about conventional language
and notation: is there a standard name or notation for the
operation which, given { a, a, a, b, c, c } and { b, b, c,
d, d } as inputs, gives as output { a, a, a, b, b, b, c,
c, c, d, d }, i.e., just adds the multiplicities? "Unions"?
"Joins"? I'm guessing "join" would be wrong because it sounds
as if it means the smallest mutliset that includes both (or all)
of the arguments as sub-multisets.
Suppose we ask how many partitions of a set collapse to
a particular partition of an integer when the members of
the set become indistinguishable. E.g., the partition
{ { 1, 3, 8}, { 2, 4, 5 }, { 6, 7 } } is one of the partitions
of the set { 1, 2, 3, 4, 5, 6, 7, 8 } that collapse to the
partition 3 + 3 + 2 of the number 8. The answer is well-known
to combinatorialists. Now generalize the problem: rather
than making ALL members of the set { 1, 2, 3, 4, 5, 6, 7, 8 }
indistinguishable, we just make some indistinguishable, so
that e.g. the set looks like { 1, 1, 1, 1, 5, 5, 7, 8 }.
Then we may ask how many partitions of the set
{ 1, 2, 3, 4, 5, 6, 7, 8 } collapse to, e.g.,
{ { 1, 1, 5}, { 1, 1, 5 }, { 7, 8 } }. The answer is
well-known (to me). I suspect the existence of about
a half-dozen people (whom I cannot name, since I have only
non-constructive evidence of their existence) who think that
_everybody_ knows the answer to this question (and I suspect
they may be -- believe-it-or-not -- mostly statisticians or
statistical genetecists). (Actually it's a fairly easy
problem; maybe they should put in on the Putnam.) This kind
of problem seems to be efficiently discussible only if one
introduces some language, and there is something to be said
for adhering to conventional nomenclature. So what is it?
-- Mike Hardy
Michael J Hardy
> A multiset is a "set with multiplicities"; each member's
> multiplicity is a positive integer. Or maybe non-negative;
> you can do it either way if you're consistent. The simplest
> natural example is the prime factorization of a natural
> number. My first question is about conventional language
> and notation: is there a standard name or notation for the
> operation which, given { a, a, a, b, c, c } and { b, b, c,
> d, d } as inputs, gives as output { a, a, a, b, b, b, c,
> c, c, d, d }, i.e., just adds the multiplicities? "Unions"?
> "Joins"? I'm guessing "join" would be wrong because it sounds
> as if it means the smallest mutliset that includes both (or all)
> of the arguments as sub-multisets.
OK, I presume the reason no one's answered is that it would
not be polite to tell me how dumb this question is, but here's
what I've decided: "union" will not do, since then the union
of two sets would not be the same as their "union" when regarded
as multisets with all multiplicities equal to 1. I've decided
on "sum". -- Mike Hardy
Dan Luecking
>Michael J Hardy (mjh...@mit.edu) wrote:
>
>> A multiset is a "set with multiplicities"; each member's
>> multiplicity is a positive integer. Or maybe non-negative;
>> you can do it either way if you're consistent. The simplest
>> natural example is the prime factorization of a natural
>> number. My first question is about conventional language
>> and notation: is there a standard name or notation for the
>> operation which, given { a, a, a, b, c, c } and { b, b, c,
>> d, d } as inputs, gives as output { a, a, a, b, b, b, c,
>> c, c, d, d }, i.e., just adds the multiplicities? "Unions"?
>> "Joins"? I'm guessing "join" would be wrong because it sounds
>> as if it means the smallest mutliset that includes both (or all)
>> of the arguments as sub-multisets.
>
>
>OK, I presume the reason no one's answered is that it would
>not be polite to tell me how dumb this question is,
For what it's worth, the holidays have kept me away.
> but here's
>what I've decided: "union" will not do, since then the union
>of two sets would not be the same as their "union" when regarded
>as multisets with all multiplicities equal to 1. I've decided
>on "sum". -- Mike Hardy
I've used "union" and didn't worry a bit about this inconsistency.
Of course I did explain the notation.
Dan
--
Dan Luecking Department of Mathematical Sciences
University of Arkansas Fayetteville, Arkansas 72701
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