You Do Need power plug adapter while traveling from Luxembourg to Saint Barthélemy.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E and in Luxembourg are C and F. This is how they look like.

Luxembourg

In Luxembourg, outlets of type C and F are in use.

Type C

Type F

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E