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We establish an improved lower bound of 10.271 for the exponential growthrate of the class of permutations avoiding the pattern 1324, and an improvedupper bound of 13.5. These results depend on a new exact structuralcharacterisation of 1324-avoiders as a subclass of an infinite staircase gridclass, together with precise asymptotics of a small domino subclass whoseenumeration we relate to West-two-stack-sortable permutations and planar maps.The bounds are established by carefully combining copies of the dominoes inparticular ways consistent with the structural characterisation. The lowerbound depends on concentration results concerning the substructure of a typicaldomino, the determination of exactly when dominoes can be combined in thefewest distinct ways, and technical analysis of the resulting generatingfunction.

A structural characterisation of Av(1324) and new bounds on its growth rate