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Assume we have a multidimensional polynomials indexed by the powers of arguments: $$P_{(i_1, i_2, \dots, i_n)}(x_1, \dots, x_n) = x_1^{i_1}x_2^{i_2}\dots x_n^{i_n}$$ I would like to find a way to index these polynomials using a single integer in a way that there are no repetitions and that we start from polynomials with smaller degree. So I am after a function $I:\mathbb{N} \rightarrow \mathbb{N}^N$ such that:

  • $P_{I(k_1)} \neq P_{I(k_2)}$ for $k_1 \neq k_2$
  • $\sum I(k_1) \leq \sum I(k_2)$ for $k_1 \leq k_2$
  • Every simple polynomial is indexed

The definition of $I$ can be recursive if its easier.

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