This image shows one solution for dealing with a staircase that is between walls for part of the stair. The chosen solution has the stair being sized width wise to accommodate the railing with balusters to be fully inside the walls at the start of the flight. The railing is centered over the stringer,  this was taken into consideration when laying out the stairs as the stair needed to be narrow enough to fit between the walls and further narrowed to allow the railing to function inside as well. The layout had to allow the  railing to overhang the stringer while still allowing a full hand grip on the railing without getting fingers pinched between the wall and the hand railing.  The railing is 3" wide and the stringer is 1 1/4" wide making the railing overhang 7/8" on each side of the stringer. One must further allow for finger room between the rail and the wall further reducing the width of the stair within the opening.  Normally when a housed stringer is used, it will "hook" up over top of the main floor in this instance however the nosing was to go right a across and the standard hook was removed to facilitate this. Note the railing was returned into the wall at the top.

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Image id: Pickoak2
Railing: #1L
Balusters: turned 1 3/4" Colonial - oak
Newel Posts: turned 3 1/2" Colonial - oak
Stairs: Straight H2S closed rise in oak 1 1/8" tread, 1 1/4" stringer 
@ Finishing Touches -