Spiral-Permutations Of Positive Integers

[Leroy Quet]

Hopefully this will inspire positive-integer-permutation-sequence ideas:

We can place the positive integers into an infinite grid's squares (one 
integer per square) by some proceedure, then read off the integers by 
another procedure (such as simply spiralling outwards 
{Ulam-spiral-style}).
In this manner we can get a large number of ways to form permutations of 
the positive integers.


Examples:

-

In the sci.math thread "Filling A Grid By Stepping 1,2,3,4,..."

http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&safe=off&threadm=b4be2fd
f.0401281733.658decab%40posting.google.com&rnum=1&prev=

I mention the problem of visiting every lattice point of an infinite 
lattice by jumping an exact number of positions which increases by one 
each move, and by visiting each lattice-point exactly once. So, in other 
words, the integer k is (k-1) steps (horizontally or vertically) from the 
integer (k-1).

Now, there are many ways this can be done, but I gave an "algorithm" 
which (hopefully) succeeded in visiting every lattice point exactly once.

I copy/paste my answer below:

>I show all positive integers up to 11 and every even up to 64 in my answer:
>  
>
>
>      46 48 50
>40 42 44  3 52 54
>38 12 14 16 18 56  
>36 10  1  2 20 58 
>34  8  6  4 22 60  *  5  7
>32 30 28 26 24 62
>    *    <-... 64
>    *
>    *
>    *
>    *
>    *
>    9
>   11
>
>(Use fixed-width font to view.)
>
>
>So, the 17,15,13 [as from left to right] are left of this image,
>off-screen, in a horizontal line and level with the row
>which reads 38, 12, 14, 16, 18, 56.
>
>Basically, the even integers spiral outwards,
>going around any lines of odd integers placed there earlier.
>
>And the odd integers form (ever lengthening) vertical
>and horizontal line-segments which line up with
>columns/rows of the even integers they are adjacent to numerically.


Now, once the grid is filled with integers, we can read off the integers 
in a simple spiral manner into a one-dimensional sequence.

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,...

We could have spiralled counter-clockwise as well, or we could have 
started at another integer, etc.

-
So, another simple example:

Write the integers in a clockwise spiral, then read off the integers in a 
counterclockwise  spiral which is centered one integer to right of first 
spiral's center.

21 22 23 24 25
20 7  8  9  10
19 6  1  2  11
18 5  4  3  12
17 16 15 14 13

permutation:
2, 1, 4, 3, 12, 11, 10, 9, 8, 7, 6, 5, 16, 15, 14, 13,...

(We must also specify that the spirals both start by going downwards, one 
to the left and one to the right.)

--

(All of the above seems to me to be more fun than mathematically useful.)

 
thanks,
Leroy Quet