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I have come up with an injective multivariate function that puts out a unique value for every configuration of four positive natural numbers provided that $\omega\ge\psi\ge\chi\ge\theta\ge1$

$f(\omega,\psi,\chi,\theta)= \frac{\omega^4+2\omega^3-\omega^2-2\omega}{24}+\frac{\psi^3-\psi}{6}+\frac{\chi^2-\chi}{2}+\theta$

f(4,2,2,1) = 18

PC1
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Leaven
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1 Answers1

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If $f$ is really injective with your restriction $\omega\ge\psi\ge\chi\ge\theta$ then it means that $f(\omega+1,1,1,1)>f(\omega,\omega,\omega,\omega)$, else we have a contradiction. So it's simple to find the value of $\omega$, by finding the largest integer $\omega$ so $f(\omega,1,1,1)$ doesn't exceed the value of the function to be inverted.

You can then find the next values in chain ($\psi$,$\chi$,$\theta$), using a similar technique.

PC1
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  • Sorry, I should've mentioned that the four numbers can only be natural numbers, we can't put 0 in any of them so the function is truly injective with that restriction. – Leaven Jun 27 '22 at 04:16
  • @leaven this doesn't change the argument. I changed the answer anyway for clarity. – PC1 Jun 27 '22 at 04:18
  • Thanks. I understand it. – Leaven Jun 27 '22 at 04:26