Largest bipartite sub-matchings of a random ordered matching or a problem with socks

Let M be an ordered matching of size n, that is, a partition of the set r2ns into 2-element subsets.The sock number of M is the maximum size of a sub-matching of M in which all left-ends of the edges precede all the right-ends (such matchings are also called bipartite).The name of this parameter comes from an amusing "real-life" problem posed by Bosek, concerning an on-line pairing of randomly picked socks from a drying machine.Answering one of Bosek's questions we prove that the sock number of a random matching of size n is asymptotically equal to n{2.Moreover, we prove that the expected average number of socks waiting for their match during the whole process is equal to 2n`1 6 .Analogous results are obtained if socks come not in pairs, but in sets of size r ě 2, which corresponds to a similar problem for random ordered r-matchings.We also attempt to enumerate matchings with a given sock number.