  We establish an improved lower bound of 10.271 for the exponential growth
rate of the class of permutations avoiding the pattern 1324, and an improved
upper bound of 13.5. These results depend on a new exact structural
characterisation of 1324-avoiders as a subclass of an infinite staircase grid
class, together with precise asymptotics of a small domino subclass whose
enumeration we relate to West-two-stack-sortable permutations and planar maps.
The bounds are established by carefully combining copies of the dominoes in
particular ways consistent with the structural characterisation. The lower
bound depends on concentration results concerning the substructure of a typical
domino, the determination of exactly when dominoes can be combined in the
fewest distinct ways, and technical analysis of the resulting generating
function.
