  We present two algorithms to list certain classes of monomino-domino
coverings which conform to the \emph{tatami} restriction; no four tiles meet.
Our methods exploit structural features of tatami coverings in order to create
the lists in $O(1)$ time per covering. This is faster than known methods for
generating certain classes of matchings in bipartite graphs.
  We discuss tatami coverings of $n\times n$ grids with $n$ monominoes and $v$
vertical dominoes, as well as tatami coverings of a two-way infinitely-wide
strip of constant height, subject to the constraint that they have a finite
number of non-trivial structural "features".
  These two classes are representative of two differing structural
characterisations of tatami coverings which may be adapted to count other
classes of tatami coverings or locally restricted matchings, such as tatami
coverings of rectangles.
