  An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with
an ordering on its vertices. These objects can be viewed as natural
generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is
well-developed and has connections and applications to extremal and enumerative
combinatorics, probability, and geometry. On the other hand, in the case $r \ge
3$ much less is known, largely due to a lack of powerful bijective tools.
Recently, Dudek, Grytczuk and Ruci\'nski made some first steps towards a
general theory of ordered $r$-matchings, and in this paper we substantially
improve several of their results and introduce some new directions of study.
Many intriguing open questions remain.
