Here is a further breakdown of the numbers of total solutions for the positions of the '+' pentomino when it is not touching the bottom boundary.

'+' pentomino location 1.

Type 5 var:'U'
299 Solutions

Type 5 var:'L'
143 Solutions

Both Variations
342 Total

'+' pentomino location 2

Type 1
13 Solutions

Type 2
39 Solutions

Type 3
22 Solutions

Type 4
19 Solutions

Type 5
169 Solutions

All Five Types
262 Total

'+' pentomino location 3.

Type 1
32 Solutions

Type 2
124 Solutions

Type 3
21 Solutions

Type 4
45 Solutions

Type 5
54 Solutions

All Five Types
276 Total

'+' pentomino location 4.

Type 1
18 Solutions

Type 2
74 Solutions

Type 3
133 Solutions

Type 4
23 Solutions

Type 5
89 Solutions

All Five Types
337 Total