À l’armoricaine is French for "in the style of Armorica." With a velouté sauce with prawns, garlic, tomatoes and oil. Armoricaine is thought to be a corruption of américaine. The origin of homard à l’américaine is controversial, some claiming that it was invented by the chef Pierre Fraisse, a native of Languedoc who spent some time in United States. The one thing that is certain is that, containing such ingredients as oil and garlic, it is unlikely to originate from Armorica (the ancient name for Brittany).
À l’arpajonnaise is French for "in the style of Arpajon." A dish containing haricot beans (US: navy beans), named after Arpajon in the Île-de-France, known for its beans.
À l’artésienne is French for "in the style of Artois." This generally indicates a dish cooked with beer.
"In the style of Saint-Germain" in the Île-de-France. Saint-Germain was once famous for peas, so this indicates the presence of peas, either fresh or dried. More uncommonly it denotes the presence of artichokes.