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

franktaw at netscape.net franktaw at netscape.net
Thu Nov 24 22:47:55 CET 2011

Starting at row 3 or later, you can only go one row down.

The differences of a graceful permutation of length n >= 3 must have 
n-1 next to n-2; n-1 must be the difference of n and 1, and n-2 is 
either the difference of n and 2 or of n-1 and 1; in either case, it 
shares a term with n-1 and thus must be next to it.

Now in the graceful permutation, below (n-1 n-2) we must have either (1 
n 2) or (n 1 n-1); as long as n >= 4, neither allows n next to n-1, so 
you can't go down another step.

Franklin T. Adams-Watters

-----Original Message-----
From: Alonso Del Arte <alonso.delarte at gmail.com>

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 
row contains the numbers from 1 to n, row n – 1 consists of the 
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 
1. For row 2 there are only two possibilities, and for row 3 there are 
four possibilities. Obviously the number of possibilities for such a
triangle that successfully goes to row 4 is finite and in fact quite 
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 
triangle infinite by starting at row 1, it does not necessarily follow 
an infinite triangle is impossible starting at some other row. And, if 
a triangle is possible, starting at say, row 5, and someone discovers 
how should he or she send it in to the OEIS? With the the first four 
omitted? Or with the first four rows put in even though they don't have 
same property as all the other rows?

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


Alonso del Arte
Author at 
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>


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