Should I include or not the days a strategy has no open positions (thus no returns) in the Sharpe ratio calculation?
No, don't include them. Otherwise you'll just wind-up with zero-value returns (or worse, forward-filled returns), which will make your Sharpe ratio reflect a performance that didn't actually occur.
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If you really think about the actual meaning of Sharpe ratios then you should come to the right conclusion yourself:
In that you obviously only want to calculate actual returns. You do not have any actual returns on days with no open positions. Hence, why would you want to include such days?
=> The core thought here is to only measure the nature of the returns not whether returns occurred or not.