  P\'olya urns are urns where at each unit of time a ball is drawn and replaced
with some other balls according to its colour. We introduce a more general
model: the replacement rule depends on the colour of the drawn ball and the
value of the time ($\operatorname{mod} p$). We extend the work of Flajolet et
al. on P\'olya urns: the generating function encoding the evolution of the urn
is studied by methods of analytic combinatorics. We show that the initial
partial differential equations lead to ordinary linear differential equations
which are related to hypergeometric functions (giving the exact state of the
urns at time n). When the time goes to infinity, we prove that these periodic
P\'olya urns have asymptotic fluctuations which are described by a product of
generalized gamma distributions. With the additional help of what we call the
density method (a method which offers access to enumeration and random
generation of poset structures), we prove that the law of the south-east corner
of a triangular Young tableau follows asymptotically a product of generalized
gamma distributions. This allows us to tackle some questions related to the
continuous limit of large random Young tableaux and links with random surfaces.
