Given the permutation $\sigma = (\sigma_{1}, \sigma_{2}, \sigma_{3}, \sigma_{4}, \sigma_{5}) = (4, 3, 5, 1, 2)$, this maps $1 \mapsto 4$, $2 \mapsto 3$, $3 \mapsto 5$, $4 \mapsto 1$, and $5 \mapsto 2$. This has the following matching diagram (and below its corresponding permutation graph, see here).
It is clear how the top diagram follows from the bottom. However, if the edges are the inversions of the permutation, then we have $1 < 2$ but $\sigma_{1} = 4 > 3 = \sigma_{2}$ so that $(1, 2)$ is an inversion but it is not in the matching diagram. Why is this? Am I understanding the permutation incorrectly?
I might be misunderstanding the conventions used here, but it looks to me like both the permutation graph and the matching diagram are mislabeled, or at least drawn very confusingly. If we read the top numbers of the diagram as the domain and the bottom numbers as the codomain (which seems like the natural reading to me), then the matching diagram shows $2$ being sent to the $5$th position, not the $3$rd position as it should be in $\sigma$. This would seem to be most naturally interpreted as the matching diagram and permutation graph for $\sigma^{-1}$ rather than $\sigma$ itself.
I welcome any corrections to this interpretation from someone more familiar with these conventions.