A distribution housing including an upstream portion and a widened downstream portion along the transverse axis relative to the upstream portion, in which an air inlet formed in the upstream portion is adapted to correspond to an outlet of an air conditioning apparatus, first and second outlets formed in the downstream portion are respectively adapted to correspond with a duct network for distributing blown air towards a de-icing board and side louvers, first pipes connect the inlet to the first outlets and extend between separation walls of the ducts and side walls of the housing, and a second pipe connects the inlet to the second outlet and extends at least partially between the duct separation walls. The second pipe opens, on the air inlet side, at the center of the air inlet, and the first pipes open on both sides of the first pipe. The housing has a different profile along the plane between the first and second pipes.