  We generalize several classical results about Schur functions to the family
of cylindric Schur functions. First, we give a combinatorial proof of a
Murnaghan--Nakayama formula for expanding cylindric Schur functions in the
power-sum basis. We also explore some cases where this formula is
cancellation-free.
  The second result is polynomiality of Kostka coefficients associated with
stretched row-flagged skew Schur functions. This implies polynomiality of
stretched cylindric Kostka coefficients. This generalizes a result by E.
Rassart from 2004.
  Finally, we also show the saturation property for the row-flagged skew Kostka
coefficients which also implies the saturation property for cylindric Schur
functions.
