SQL Server Standard edition attach partitioned database
You cannot restore a database that contains partitioned tables in Standard Edition. You also cannot attach a database without supplying all the data files. You could restore individual files or filegroups, but it won't achieve what you're after. 

Your only option is to get rid of partitioning before you take a backup and move it to Standard Edition.

BTW, if you query sys.dm_db_persisted_sku_features you'll get a list of all the active features in the database that might impact your migration.

EDIT 2017-09-15: not true any more since SQL Server 2016 SP1.