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I am currently misunderstanding a very basic section of a paper I am reading. An extract follows:

With $\mathcal{L} = \{v_1 = mild, v_2 = severe, v_3 = very~mild, v_4 = very~severe\}$. In this case, there is a permutation $\sigma$ of $1,\ldots ,L$ such that $v_{\sigma(1)},\ldots,v_{\sigma(L)}$ is a sequence of labels correctly ordered according to their semantic meaning. Two valid permutations are then $σ_1 = (2, 3, 1, 4)$ and $σ_2 = (3, 2, 4, 1)$ that rank the labels from $very~mild$ to $very~severe$ and from $very~severe$ to $very~mild$, respectively.

But this is where I am misunderstanding something. For me $\sigma_1$ should be equal to $(3,1,2,4)$ to correspond to the semantic ranking $very~mild$ to $very~severe$. And $\sigma_2$ should be $(4,2,1,3)$.

Can someone explain to me the obvious error I am making in my deductions?

Astrid
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    The notation for permutations is notoriously non-uniform. The paper uses $\sigma = (a,b,c,d)$ to say that $v_1$ goes to place $a$, $v_2$ to place $b$ etc. You use it to say that $v_a$ goes to place $1$, $v_b$ to place $2$ etc. – Daniel Fischer Nov 14 '17 at 14:42
  • Aha, was not aware of that. Thanks – Astrid Nov 14 '17 at 15:06

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