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?
