  Suppose $\lambda$ and $\mu$ are integer partitions with
$\lambda\supseteq\mu$. Kenyon and Wilson have introduced the notion of a
cover-inclusive Dyck tiling of the skew Young diagram $\lambda\setminus\mu$,
which has applications in the study of double-dimer models. We examine these
tilings in more detail, giving various equivalent conditions and then proving a
recurrence which we use to show that the entries of the transition matrix
between two bases for a certain permutation module for the symmetric group are
given by counting cover-inclusive Dyck tilings. We go on to consider the
inverse of this matrix, showing that its entries are determined by what we call
cover-expansive Dyck tilings. The fact that these two matrices are mutual
inverses allows us to recover the main result of Kenyon and Wilson.
  We then discuss the connections with recent results of Kim et al, who give, a
simple expression for the sum, over all $\mu$, of the number of cover-inclusive
Dyck tilings of $\lambda\setminus\mu$. Our results provide a new proof of this
result. Finally, we show how to use our results to obtain simpler expressions
for the homogeneous Garnir relations for the universal Specht modules
introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke
algebras.
