Determine the parity and the inverse for each of the following permutation of {1, 2, . . . , 9}:

Question

So i was given this question. Determine the parity and the inverse for each of the following permutation of {1, 2, . . . , 9}:

(a) (987654321)

(b) (135792468)

I don't understand how to go about this. I checked this example Parity and Inverse of Permutations (Odd and Even) but i did not understand how they solved it. They took the third example and solved the rest.

Answer 1

To find the inverse permutation, write the permutation $\sigma \in S_9$ as a list, i.e. \begin{array}{c||cc} i & 1 & \cdots & 9\\ \hline \sigma(i) & \sigma(1) & \cdots & \sigma(9) \end{array}

Then flipping the top and bottom rows will give you the inverse permutation \begin{array}{c||cc} i & \sigma(1) & \cdots & \sigma(9)\\ \hline \sigma^{-1}(i) & 1 & \cdots & 9 \end{array}

To compute the sign of the permutation, you can use the symbolic formula $$ (-1)^\sigma = \prod_{1 \leq i < j \leq 9} \frac{x_{\sigma(i)} - x_{\sigma(j)}}{x_i - x_j}, $$ where $x_1, \dots, x_9$ are just dummy variables, and each monomial $(x_p - x_q)$ in the numerator will cancel with a corresponding monomial $(x_p - x_q)$ or $(x_q - x_p)$ in the denominator [why?], leaving you with an answer of $\pm 1$.