# [seqfan] Re: Inequivalent matrices

Neil Sloane njasloane at gmail.com
Fri Apr 29 02:45:54 CEST 2022

```> Maybe we should leave a map to future generations

That is exactly what the Index to the OEIS is for.

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

On Wed, Apr 27, 2022 at 11:48 PM M. F. Hasler <oeis at hasler.fr> wrote:

> On Wed, Apr 27, 2022 at 10:42 PM Brendan McKay via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
> > A180985 (rows and columns in lexicographic order) is different
> > from A002724 (equivalence classes under row and column permutation).
>
> Equivalence classes under row and column permutation for Z/Z3 is A052269.
> > For Z/Z4 it is A052271, for Z/Z5 it is A052272, and for Z/Z6 it is
> A246112.
>
> All of these except Z/Z6 have formulas provided by Christian Bower.
> > A246106   is a 2-D version.  [MH: i.e., column k  for entries in Z/kZ ]
> >
>
> Great!  Thank you very much!
> Some further refs :
> A002724 (n X n, Z/2Z)  is the diagonal of  oeis.org/A028657 (for n x k
> rectangular {0,1}-matrices)
>
> I tried to figure out the distinction between
> oeis.org/A246106  and oeis.org/A256069, which seems to have the same
> definition
> (except for the layout : triangle vs square matrix), but different
> numbers...
> Do I guess correctly that A256069 must have each of {1...k} present at
> least once somewhere,
> while A246106 may use only a subset of all the possible coefficients?
> It's interesting (not totally obvious to me, a priori) that they are
> related by such a simple binomial transform.
>
> Anyway, thanks a lot again for your help in this "treasure hunt"!
> Maybe we should leave a map to future generations... :-)
>
> - Maximilian
>
> On 28/4/2022 4:52 am, M. F. Hasler wrote:
> > > Dear sequence fans,
> > >
> > > I was looking for the number of inequivalent matrices with entries in
> > Z/mZ
> > > (or {0,...,m-1}, or 1...m), modulo permutations or rows and columns.
> > > I think for m=2 this is given in Ron Hardin's oeis.org/A180985
> > > (although I may have overlooked something, and the definition using
> > > lexicographic monotonicity of rows and columns might not be completely
> > > equivalent to having a canonical representative of a class modulo the
> > > permutations I mentioned...?).
> > > It was easy to write a program that constructs the inequivalent
> > > {0,1}-matrices explicitly to enumerate them, and it should not be too
> > > difficult to find a recursive formula for their number.
> > >
> > > But the case m>2 (e g., "ternary" matrices etc.) seems slightly more
> > > complicated.
> > > Did someone already investigate on this?
> > > (The requirement of lexicographic order of rows and columns seems no
> more
> > > sufficient, since e.g., the 2×2  ternary matrix
> > > 02
> > > 10 would satisfy this, but it is equivalent to the matrix
> > > 01
> > > 20. How can the former be "excluded" in a simple manner?
> > >   I wrote a program that gives the canonical (lexicographically
> smallest)
> > > representative of any matrix, and it recognizes the equivalence in the
> > > above case, but I have not yet fixed my enumeration program so that it
> > > wouldn't even "produce" the former candidate.
> > (...)
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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