The firing order makes no difference in the "cycle" of the engine. A gas engine is most efficient when the "compression" stroke is 7 - 8 to one and the "expansion" (power) stroke is in the 13 - 14 to 1 range

A standard "Otto" engine has equal length compression and power stokes. In most engines this ranges from 8.0 to 1 to 10.5 to 1 compression and expansion ratios.

The true "Atkinson" cycle engine increases the length of the power stroke in relation to the compression stroke. A modern adaptation of this is the "Miller" method to achieve this variation. Although still called an "Atkinson" cycle, the static compression of the engine is increased to 13 to 1 or so and the intake valve is kept open later in the compression cycle.

This "bleeds off" some of the charge to be compressed giving the engine an effective 8.5 to 1 (roughly) compression ratio and a 13 to 1 (again roughly) expansion ratio. The engine will not produce as much power using this method (less air-fuel in the cylinder) but it is very efficient.