  Preference orderings are orderings of a set of items according to the
preferences (of judges). Such orderings arise in a variety of domains,
including group decision making, consumer marketing, voting and machine
learning. Measuring the mutual information and extracting the common patterns
in a set of preference orderings are key to these areas. In this paper we deal
with the representation of sets of preference orderings, the quantification of
the degree to which judges agree on their ordering of the items (i.e. the
concordance), and the efficient, meaningful description of such sets.
  We propose to represent the orderings in a subsequence-based feature space
and present a new algorithm to calculate the size of the set of all common
subsequences - the basis of a quantification of concordance, not only for pairs
of orderings but also for sets of orderings. The new algorithm is fast and
storage efficient with a time complexity of only $O(Nn^2)$ for the orderings of
$n$ items by $N$ judges and a space complexity of only $O(\min\{Nn,n^2\})$.
  Also, we propose to represent the set of all $N$ orderings through a smallest
set of covering preferences and present an algorithm to construct this smallest
covering set.
  The source code for the algorithms is available at
https://github.com/zhiweiuu/secs
