[seqfan] Is an infinite triangle of graceful permutations possible?

Alonso Del Arte alonso.delarte at gmail.com
Thu Nov 24 22:13:56 CET 2011

As I pondered the topic of graceful permutations, it occurred to me that it
might be possible to make a triangle of graceful permutations in which each
row contains the numbers from 1 to n, row n – 1 consists of the differences
of row n, and that row is itself a graceful permutation of the numbers 1 to
n – 1. Naturally, my first attempt started with row 1.

2 1
3 1 2
4 1 2 4—nope, doesn't work.

In fact, it is impossible to make the triangle infinite by starting at row
1. For row 2 there are only two possibilities, and for row 3 there are only
four possibilities. Obviously the number of possibilities for such a
triangle that successfully goes to row 4 is finite and in fact quite small,
and I have tried them all. Unless I have made a mistake somewhere (which is
possible, I admit), none of those triangles can be taken to row 5.

However, if I am correct in my assertion that it is impossible to make the
triangle infinite by starting at row 1, it does not necessarily follow that
an infinite triangle is impossible starting at some other row. And, if such
a triangle is possible, starting at say, row 5, and someone discovers it,
how should he or she send it in to the OEIS? With the the first four rows
omitted? Or with the first four rows put in even though they don't have the
same property as all the other rows?

By the way, Happy Thanksgiving ("Jour de l'Action de Grâces")!


Alonso del Arte
Author at SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>

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