Originally Posted by
MSPeconomist
He does need a visa if he spends too much time in the schengen zone within a certain period--IIRC the limit might be 90 days at a time and something like 90 days out of 180 if he enters without visas.
There is no such limit if the US citizen has permanent residency status -- card or not -- in the Schengen country to which they are headed when showing up at a Schengen port of entry. No visa needed.
The situation of the OP's child doesn't make sense, and it seems like something more must have been involved than a recognized permanent resident merely lacking the permanent residency card for presentation at the port of entry to the Schengen zone.