  In this paper, we introduce a class of $(P, \omega)$-partitions that we call
periodic $(P, \omega)$-partitions, then prove that such $(P,
\omega)$-partitions satisfy a homogeneous first-order matrix difference
equation. After defining an appropriate counting problem for the above $(P,
\omega)$-partitions, we show that as a consequence of this equation, periodic
$(P, \omega)$-partitions can be enumerated with constant coefficient linear
recurrence relations. By analysing the above matrix difference equation, we
also prove a result for the asymptotic growth rate for the number of periodic
$(P, \omega)$-partitions. The results of this paper generalizes and strengthens
the constant coefficient linear recurrence results proved by Sun and by
L\'opez, Mart\'inez, P\'erez, P\'erez, and Basova for enumerating standard
Young tableaux on shifted strips with constant width.
