What are the #P-hard version of Latin square and sudoku? I know that filling partial latin square and solving sudoku are NP-hard. But what are the counting versions of these two problems?
The counting problems corresponding to your two decision problems are:

Given a partially filled Latin square, in how many ways can it be completed to a Latin square?
Given a partially filled Sudoku puzzle, in how many ways can it be completed to a Sudoku solution?