5
$\begingroup$

This is a practical and very soft question, with the combinatorial database http://www.findstat.org in mind.

I have a few, around 20, families of combinatorial objects, for example Dyck paths, permutations, perfect matchings, graphs, etc., together with a few, around 200, maps between them. The maps need not be injective or surjective or have any special properties. (Except perhaps that they appear in the literature and are therefore "interesting".)

Some examples of such maps might be the reversal of a Dyck path, various classical maps that send a Dyck path to a 321-avoiding permutation, the map that sends a permutation to its shape under the Robinson-Schensted correspondence, etc.

Thus, we have (a very small) category, which I'd like to visualize.

The aspect that makes this interesting and non-trivial is, that these maps satisfies numerous identities: many maps are involutions, some are idempotent, many maps commute with other maps or are conjugate to other maps, etc. I have all these data.

So, more precisely: I'm looking for a way to visualize (graphically!) these identities.

Improve this question
$\endgroup$
9
  • $\begingroup$ Silly remark: All relations between these maps can be drawn as commutative diagrams. For commutativity relations this makes sense somehow, but for idempotent maps it is kind of "over the top". $\endgroup$
    – HeinrichD
    Dec 28, 2016 at 17:02
  • $\begingroup$ Yes. The problem is the amount of the data. So, to clarify: the question is about an economic visual representation. $\endgroup$
    – Martin Rubey
    Dec 28, 2016 at 17:16
  • $\begingroup$ It might be overkill, but you could put all the generating morphisms and relations into Globular and then play around with it. Globular is designed to manipulate higher categories, but it should work just fine for 1-categorical data. Incidentally, do you have any map which takes as input a pair of objects, like, say, a Dyck path and a permutation? This would start using some kind of monoidal structure, which is a sort of higher categorical structure. $\endgroup$
    – Tim Campion ♦
    Dec 28, 2016 at 17:52
  • $\begingroup$ @TimCampion: Except that I have no idea how to use it, sounds interesting! $\endgroup$
    – Martin Rubey
    Dec 28, 2016 at 18:48
  • $\begingroup$ @TimCampion: Could you show how to visualize the identities "reverse o inverse = inverse o complement", "reverse o complement = complement o inverse", etc.? (keeping in mind that I have a list of a few thousand such identities...) $\endgroup$
    – Martin Rubey
    Dec 28, 2016 at 18:58

1 Answer 1

Reset to default
3
$\begingroup$

Despite some interesting proposals given in the comments of the question, I finally ended up with a rather straightforward listing, mainly because the amount of data is quite large. An example can be studied at http://findstat.org/Mp00023.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.