Permutation of the integers

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A permutation of the integers is a sequence in which each of the integers (the members of 

ℤ: {..., –7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7, ...})

occurs exactly once, generally in a position other than its usual position in ascending order. The identity permutation of the integers gives the sequence of integers in ascending order, which happens to be a doubly infinite sequence (which are not admissible in the OEIS, since OEIS sequences must have an initial term).

A001057 is a very simple example of such a permutation, which gives a singly infinite sequence:

{0, 1, –1, 2, –2, 3, –3, 4, –4, 5, –5, 6, –6, 7, –7, 8, –8, 9, –9, 10, –10, 11, –11, 12, –12, 13, –13, 14, –14, 15, –15, 16, –16, 17, –17, 18, –18, 19, –19, 20, –20, 21, –21, 22, –22, 23, –23, 24, –24, ...}

Sequences like these are known to be permutations because they were so defined. Certain sequences arise in other problems and are proved to be permutations. Others are conjectured to be permutations, until a repeated or absent term can be found.

One may want to distinguish between two kinds of permutations of doubly infinite sequences:

a permutation giving another doubly infinite sequence (not admissible in the OEIS, since not well-ordered)
a permutation giving a singly infinite sequence (admissible in the OEIS, since well-ordered)