[seqfan] Re: balls of u distinct colors filling u unlabeled urns of capacity b

[Pontus von Brömssen]

It's A257463!

Cheers,

/Pontus

On Tue, Jun 20, 2023 at 4:40 AM Max Alekseyev <maxale at gmail.com> wrote:

> Hi Richard,
>
> I confirm your counts with quick'n'dirty PET implementation:
> https://sagecell.sagemath.org/?q=hpcmyk
>
> Regards,
> Max
>
>
>
> On Mon, Jun 19, 2023 at 3:26 PM Richard J. Mathar <mathar at mpia-hd.mpg.de>
> wrote:
>
> > This here is a combinatorial enumeration problem, where I'm wondering
> > whether
> > this is in the OEIS:
> > Let there be u unlabeled urns, also u colors, and b balls per color, so
> > there
> > is a total of u*b balls.
> > In the following examples colors enumerated 0,1,...,u-1.
> > "Unlabeled" urns means that permuting urns (with their content) does not
> > generate new/additional countable objects.
> > Distribute all u*b balls until each urn is filled with b balls; the
> > terminology
> > "urn" indicates that the balls of various colors inside an urn are not
> > sorted
> > and considered a multiset for our purposes. If we delimit the urns
> > by vertical bars and put the ball's colors in between,
> > these are the basic examples:
> >
> > b=2, u=1 gives 1 distinct configurations:
> > | 0 0
> >
> > b=1, u=2 gives 1 distinct configurations:
> > | 0 | 1
> >
> > b=2, u=2 gives 2 distinct configurations:
> > | 0 0| 1 1
> > | 0 1| 0 1
> >
> > b=3, u=2 gives 2 distinct configurations:
> > | 0 0 0| 1 1 1
> > | 0 0 1| 0 1 1
> >
> > b=3, u=3 gives 10 distinct configurations:
> > | 0 0 0| 1 1 1| 2 2 2
> > | 0 0 0| 1 1 2| 1 2 2
> > | 0 0 1| 0 1 1| 2 2 2
> > | 0 0 1| 0 1 2| 1 2 2
> > | 0 0 1| 0 2 2| 1 1 2
> > | 0 0 2| 0 1 1| 1 2 2
> > | 0 0 2| 0 1 2| 1 1 2
> > | 0 0 2| 0 2 2| 1 1 1
> > | 0 1 1| 0 1 2| 0 2 2
> > | 0 1 2| 0 1 2| 0 1 2
> >
> > b=4, u=2 gives 3 distinct configurations:
> > | 0 0 0 0| 1 1 1 1
> > | 0 0 0 1| 0 1 1 1
> > | 0 0 1 1| 0 0 1 1
> >
> > b=5, u=2 gives 3 distinct configurations:
> > | 0 0 0 0 0| 1 1 1 1 1
> > | 0 0 0 0 1| 0 1 1 1 1
> > | 0 0 0 1 1| 0 0 1 1 1
> >
> > b=2, u=4 gives 17 distinct configurations:
> > | 0 0| 1 1| 2 2| 3 3
> > | 0 0| 1 1| 2 3| 2 3
> > | 0 0| 1 2| 1 2| 3 3
> > | 0 0| 1 2| 1 3| 2 3
> > | 0 0| 1 3| 1 3| 2 2
> > | 0 1| 0 1| 2 2| 3 3
> > | 0 1| 0 1| 2 3| 2 3
> > | 0 1| 0 2| 1 2| 3 3
> > | 0 1| 0 2| 1 3| 2 3
> > | 0 1| 0 3| 1 2| 2 3
> > | 0 1| 0 3| 1 3| 2 2
> > | 0 2| 0 2| 1 1| 3 3
> > | 0 2| 0 2| 1 3| 1 3
> > | 0 2| 0 3| 1 1| 2 3
> > | 0 2| 0 3| 1 2| 1 3
> > | 0 3| 0 3| 1 1| 2 2
> > | 0 3| 0 3| 1 2| 1 2
> >
> > If we put these counts into a table we have
> >
> > b \ u | 1  2  3  4  5
> > -------------------
> > 1     | 1  1  1  1  1
> > 2     | 1  2  5 17 73
> > 3     | 1  2 10
> > 4     | 1  3
> > 5     | 1  3
> >
> > The basic question is: are my counts correct, and is this known?
> >
> > Remark:
> > One can also rephrase the problem as considering u tables, each with b
> > seats,
> > and u groups (enumerated 0,...,u-1) each with b members. In a
> > sort of menage problem all u*b people are seated at the tables.
> > This is related to https://math.stackexchange.com/questions/1140111 :
> > this asks for u=16, b=3 how many configurations exist where all
> > tables have only people of different groups; that would be the subset
> > of our cases where all integers between the bars are required to be
> > distinct.
> >
> >
> > Remark:
> > If we consider distinguishable/labeled urns, we end up at A257493.
> > The integer matrices in A257493 would indicate in rows labeled
> > by color how many balls of that color end up in which urn (=labeled
> > column).
> >
> > Best Regards,
> > Richard
> >
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> >
>
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