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

The correct option is Had, would.

Let us analyze the grammatical structure of the sentence step-by-step to understand why this option is correct.

1. Analyzing the First Blank: Conditional Clause
The second part of the sentence uses the past conditional structure: "I wouldn’t have had...". This indicates a hypothetical condition in the past (Third Conditional).
In a third conditional sentence, the condition ("if" clause) is typically in the past perfect tense: If I had permitted him to leave...
When "if" is omitted to form an inverted conditional structure for emphasis, the auxiliary verb Had is placed at the beginning of the clause:
Had I permitted him to leave... (which means "If I had permitted him to leave...")
Therefore, the first blank must be filled with Had.

2. Analyzing the Second Blank: Question Tag
The statement prior to the question tag is: "...I wouldn’t have had any problem with him being absent..."
In English grammar, a question tag is formed using the auxiliary verb from the main clause. If the main statement is negative ("wouldn't" / "would not"), the question tag must be positive ("would").
Since the main clause helper is "wouldn't" (negative), the tag must be would (positive) followed by the pronoun "I".
Thus, the question tag is: "...would I?"

Combining these two analyses, the correct words to fill the blanks are Had and would, making the complete sentence:
"Had I permitted him to leave, I wouldn’t have had any problem with him being absent, would I?"