[seqfan] Re: Inequivalent matrices

M. F. Hasler oeis at hasler.fr
Thu Apr 28 05:48:11 CEST 2022

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
(except for the layout : triangle vs square matrix), but different
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.
> (...)

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