-------------------I permitted him to leave, I wouldn’t have had any problem with him being absent, --------------1?

The correct answer is Had, would.

Let's analyze the sentence structure and the rules of conditional clauses to understand why this option is correct.

Step 1: Analyze the first clause
The sentence begins with: "-------------------I permitted him to leave, I wouldn’t have had any problem with him being absent..."
The main clause contains "I wouldn't have had...", which is in the conditional perfect form (would + not + have + past participle). This indicates a third conditional sentence structure, which is used to talk about hypothetical or counterfactual situations in the past.
In a standard third conditional sentence, the "if-clause" uses the past perfect tense: "If I had permitted him to leave..."
However, we can omit "if" by inverting the subject and the auxiliary verb "had". This is known as conditional inversion:
"Had I permitted him to leave..." (which means "If I had permitted him to leave...").
Therefore, the first blank must be filled with Had.

Step 2: Analyze the second clause (the question tag)
The sentence ends with a question tag: "--------------1?"
A question tag is formed using the auxiliary verb from the main clause and the pronoun corresponding to the subject.
Here, the main clause is: "I wouldn’t have had any problem with him being absent..."
The auxiliary verb in the main clause is wouldn't (which is negative).
According to the rules of question tags, a negative statement takes a positive question tag. Therefore, the negative "wouldn't" becomes the positive would in the tag.
Combined with the subject pronoun "I", the question tag is "would I?".
Thus, the second blank must be filled with would.

Conclusion:
Combining the two parts, we get Had for the first blank and would for the second blank, making the complete sentence:
"Had I permitted him to leave, I wouldn’t have had any problem with him being absent, would I?"