The burden of taxation is shared among suppliers and demanders according to the price elasticities of supply and demand. The more elastic side carries less of the tax burden.
To understand this, note that the tax effectively increases the price demanders pay and decreases the price suppliers get. Elasticity tells us how demanders and suppliers react to this price change. For demanders this means buying less and for suppliers it means supplying less. If you are more elastic then you react to the tax more (by definition of elasticity), which means you avoid it more.
To illustrate this, imagine a tax on cigarettes. We assume that smokers who are addicted will have to buy cigarettes at any price (as they are addicted). This means they do not react and have 0 elasticity of demand. In that case, imposing a tax on cigarettes (a price increase) will not affect how many cigarettes are bought. So the producers of cigarettes are making just as much profit as before, while the consumers are paying higher prices, hence the consumers carry the full burden of the tax.
You could imagine a similar argument, where the supplier produces less because of the tax, thereby causing prices to increase and hurting the consumer.
In your case the producer is totally inelastic, while consumers have some elasticity and hence carries the burden of the tax (as he cannot react and reduce the number of seats).
Alternatively imagine a tax put on producers, where the producers are inelastic, but consumers are elastic. The producers when trying to push the tax onto the consumers (raise the price) the consumers will buy less (are elastic) and thus reduce the suppliers profits, so the producer cannot push the full tax onto the consumers if he's maximizing profits. He can only push a part of it, which means his profits will partially fall and buyers will pay higher prices. Hence both carry the tax burden. The producer can push less of the tax increase the more elastic the consumers are compared to him.
We can also see this general point graphically. The figures I have for this are when the tax is on the producers so affects the supply, but the result is equivalent to a tax on the consumers affecting demand. The blue region represents the surplus lost by the consumer that is captured as tax revenue as a result of the tax and the red region represents the surplus lost by the producer that is captured as tax revenue as a result of the tax. The grey region is deadweight loss resulting from the tax -- it is surplus lost, but not captured as tax revenue. In the first plot below, supply is relatively inelastic (i.e. not very steep) and demand is quite elastic (i.e. steep). In the second, supply is relatively elastic (i.e. steep) and demand is quite inelastic (i.e. not very steep).