  Birational rowmotion is an action on the space of assignments of rational
functions to the elements of a finite partially-ordered set (poset). It is
lifted from the well-studied rowmotion map on order ideals (equivariantly on
antichains) of a poset $P$, which when iterated on special posets, has
unexpectedly nice properties in terms of periodicity, cyclic sieving, and
homomesy (statistics whose averages over each orbit are constant) [AST11, BW74,
CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context,
rowmotion appears to be related to Auslander-Reiten translation on certain
quivers, and birational rowmotion to $Y$-systems of type $A_m \times A_n$
described in Zamolodchikov periodicity.
  We give a formula in terms of families of non-intersecting lattice paths for
iterated actions of the birational rowmotion map on a product of two chains.
This allows us to give a much simpler direct proof of the key fact that the
period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$
(first proved by D.~Grinberg and the second author), as well as the first proof
of the birational analogue of homomesy along files for such posets.
