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Birational rowmotion is an action on the space of assignments of rationalfunctions to the elements of a finite partially-ordered set (poset). It islifted from the well-studied rowmotion map on order ideals (equivariantly onantichains) of a poset $P$, which when iterated on special posets, hasunexpectedly nice properties in terms of periodicity, cyclic sieving, andhomomesy (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 certainquivers, 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 foriterated 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 theperiod 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 proofof the birational analogue of homomesy along files for such posets.

Paths to Understanding Birational Rowmotion on Products of Two Chains