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Background

In this thread here at MathStackExchange

"N-free" permutations of (0,...,n-1) and pairs of orthogonal vectors in n-space

Paul Sinclair kindly took the time to discuss with me a certain linear-algebraic result obtained by John Burkardt.

This discussion has led me to reformulate my original question as a new question involving the interpretation of a permutation of 0,,,n-1 as a set of n 2-dimensional vectors rather than a set of two n-dimensional vectors.

In particular, if an "N-free" permutation P of 0,...n-1 is interpreted as a set of n 2-dimensional vectors in the obvious way, then the relationship between P and an "N-free" poset Po can readily be seen from the simple diagrams of a language-theoretic derivation tree and corresponding labelled bracketting at the bottom of this page:

https://www.thesyntacticretina.com/

Further, this relationship is important because it allows the extension of certain long-standing results on "N-free" permutations to "N-free" posets, which have been extensively studied in their own right independently of N-free permutations, e.g. here:

http://www.sciencedirect.com/science/article/pii/0166218X85900307

(Note, however, that the extension of results on permutations to results on posets involves more effort than one might think, because although this extension is straightforward, it requires a fair amount of mechanics to do correctly.)

Question:

Are there any OTHER known mathematical contexts in which it is useful to think of a permutation 0,...,n-1 as a set of n 2-dimensional vectors rather than as a set of 2 n-dimensional vectors (APART from the context mentioned above involving permutations and posets?)

David Halitsky
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  • @PaulSinclair hope the question above makes sense to you! – David Halitsky Nov 25 '17 at 15:21
  • Did you notice that "thesyntacticretina" contains "cretin" as a substring? For a more embarrassing example of this, see http://www.bbc.co.uk/bbcthree/item/cbd0fd47-3f92-494e-8154-da08a565498e – Angina Seng Nov 25 '17 at 15:33
  • LOL! - that was funny, even if it WAS an intentional slam !!! Thanks for the laugh ! – David Halitsky Nov 25 '17 at 15:36
  • Could you please expand on "the obvious way" to interpet "0,,,n-1" as a set of $n$ 2-dimensional vectors? Did you mean $2n-1$ every time you write $n$, or does the three-commas-in-a-row have a particular meaning to you that is different from simply $0,1,\ldots,n-1$? – hmakholm left over Monica Nov 25 '17 at 18:40
  • I am very sorry that I've used confusing notation. The permutation (051423) of (012345) creates the six ordered pairs (0,0),(1,5),(2,1), (3,4), (4,2),(5,3), each of which can be interpreted as a radius vector relative to the usual OXY coordinate system in the plane. This is shown visually in the diagram at the bottom of the page I linked to: https://www.thesyntacticretina.com/. Thank you very much for taking the time to ask for clarification. – David Halitsky Nov 25 '17 at 18:50
  • In serious mathematics, I can't think of even one example where considering permutations as vectors was useful. Playing around like this for fun, sure - you do all sorts of stuff that otherwise would be nonsensical. Though I suppose this statement depends on what is considered "serious" mathematics and it certainly depends on what I've personally encountered and remember. – Paul Sinclair Nov 25 '17 at 20:44
  • Thanks Paul. Because there is a result from the 1980's on N-free permutations, and because David Wagner (Waterloo) extended this result to N-free posets in 1995, I wanted to be sure that: i) no one recalls any related work since then; ii) David did not overlook any work done prior to 1995 Given your breadth and depth, I think I can safely assume that no such work exists. – David Halitsky Nov 25 '17 at 20:49
  • I should also add this paper by Robert Jamison (Clemson) as another example of "known" work requiring citation: https://books.google.com/books?id=Q81Opo630HkC&pg=PA328&lpg=PA328&dq=jamison+dna+graphs&source=bl&ots=QiunZBpO9A&sig=EMJj0dpBnz4aVHT038pkMbHRqzg&hl=en&sa=X&ved=0ahUKEwiSlrD7ztrXAhWjYt8KHQkUCfMQ6AEIPjAE#v=onepage&q=jamison%20dna%20graphs&f=false – David Halitsky Nov 25 '17 at 21:01
  • The usual presentation of permutations considers a permutation to be a bijective function from ${1,\ldots,n}$ to itself -- and given the usual set-theoretic modeling of functions, this is exactly a set of $n$ ordered pairs. But I'm not aware of any particular context where viewing those ordered pairs as vectors is productive. – hmakholm left over Monica Nov 25 '17 at 22:28
  • Thanks again @HenningMakholm for taking the time to respond. It's good that you've concurred with Paul's opinion that there exists no substantive prior work which interprets the ordered pairs of a permutation as 2-dimensional vectors, because this concurrence makes it much less likely that either of you is offering an idiosyncratic opinion which might well be incorrect. At the same time, I hope that neither of you would go so far as to say that because the vector intepretation has not been used productively in the past, then perforce it CANNOT be so used. – David Halitsky Nov 26 '17 at 03:40
  • @PaulSinclair - please see my last comment above to Henning – David Halitsky Nov 26 '17 at 03:42
  • "Given your breadth and depth, I think I can safely assume that no such work exists." Thanks for the confidence, but it is definitely misplaced in me. My doctorate was in differential geometry, and as I've worked in industry for the last 30 years, I am seriously out of touch with research in even that field, much less others. – Paul Sinclair Nov 26 '17 at 04:02
  • @PaulSinclair - if your doctorate was in differential geometry, you might be interested in the question I just posted here: https://math.stackexchange.com/questions/2537513/suppose-it-can-be-shown-that-any-rightleft-regular-finite-state-grammar-implic (or perhaps not!) – David Halitsky Nov 26 '17 at 04:21
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    Your questions need us to read 3 other questions plus ... You are supposed to make them as simple, clear and self-contained as possible. It seems nobody understood what you meant with "2-dimensional vectors representing permutations" – reuns Nov 26 '17 at 04:48
  • @reuns - I am sorry I've apparently misunderstood the site's intstructions/suggestions. I thought that I was supposed to provide "background" for any given question. In this case, the "background" for each successive question happens to be discussion resulting from a prior question. – David Halitsky Nov 26 '17 at 04:53

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