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?