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For which pairs of integers $0, is there a function $f:\mathbb N^i \to \mathbb N$, $f(x_1,x_2,...,x_i)$ which outputs every positive integer exactly once when any $i-j$ of the variables are kept constant on any $(i-j)$-tuple of positive integers, while the other $j$ variables are varied over all $j$-tuples of positive integers?

And is there a $g$, defined on all infinite sequences of positive integers, such that for all such sequences, if we replace any one element by a variable, say $x$, then $g(x)$ is a bijection between the integers? (ie. $j=1, i=\infty$)

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    There is always such a function if $j=1$ and $i$ is finite: take the bitwise exclusive-or. (With the convention that $\mathbb{N}$ contains 0 - obviously this choice doesn't affect the question.)2012-01-06

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