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QUESTION Count the number of inversions in each of the following
permutations:

\begin{description}

\item[(a)]
1342

\item[(b)]
34215

\item[(c)]
635241

\end{description}


ANSWER An inversion in a permutation
$\sigma_1,\sigma_2,\ldots,\sigma_n$ is a pair
$(\sigma_r,\sigma_s)$ with $\sigma_r>\sigma_s$ but $r<s$.

The question can be solved either by counting such pairs or by
drawing pictures and counting crossings.

\begin{description}

\item[(a)]
Two inversions: (3,2),(4,2).

\item[(b)]
Five inversions: (3,2),(3,1),(4,2),(4,1),(2,1).

\item[(c)]
Twelve inversions:

(6,3),(6,5),(6,2),(6,4),(6,1),(3,2),

(3,1),(5,2),(5,4),(5,1),(2,1),(4,1).

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