Hi Mike: Perhaps I've developed gamma disease (gamma = ! except off by one). I'll have to see more data to see which definition works out better. At 09:03 AM 4/17/2018, Mike Speciner wrote:
If d=0 is the thing that moves the first element to the front, i.e., a no-op, then shouldn't d=5 move the sixth element (i.e., 5) to the front?
On 17-Apr-18 10:33, Henry Baker wrote:
Has anyone studied the following types of sequences as *purely mathematical* objects?
Start with an infinite sequence of non-negative integers in their natural order:
0,1,2,3,4,5,6,...
Let me call this sequence the "identity sequence".
Given a non-negative integer d, *permute* this infinite sequence by moving the d'th element to the front. Thus, given 5, we get
4,0,1,2,3,5,6,...
Notice that when d=0, nothing changes, so it is the identity.
We can define -d to be the *inverse* permutation to +d.
We can now define a *sequence* of integers operating on our initial sequence; we call these sequences of "d" values, "d-sequences".
Thus, the d-sequence 5,-5 produces the original sequence, as does -5,5.
Any 0,0,0 subsequence can be eliminated from a d-sequence, as it doesn't do anything.
Here's the result of the first 10 digits of pi operating as a d-sequence on the identity sequence:
4,2,1,8,3,0,5,6,7,9,10,11,...
We note that when d operates on a sequence, the elements indexed from d and greater don't change; i.e., only the elements indexed by <d are permuted.
Here are some obvious properties of d-sequences:
The d-sequence 0,1,2,3,...,n *reverses* the first n elements of the original sequence.
The d-sequence k^n cyclically permutes the first k elements by n; thus k^k is the identity.
Here are some obvious questions:
* Are these d-sequences related to some other types of sequences? Perhaps this definition of d-sequences isn't the most elegant?
* Clearly -reverse(seq) cancels seq; what about things like palindromes?
* Consider d-sequences *sums*. Any interesting properties?