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Let's try: ! is the negation operator, _ is an array name (it could have been "array", "pippo" or anything else, but undoubtedly a simple underscore is minimal and effective).
The expression _[$1]++ increments the value of the element of the array with index $1 by one. Since the ++ comes after, the expression is evaluated before the increment so that at the first occurrence of $1 the value is still 0 (false). For any other occurrence of $1 the value is >0 (true).
The negation invert the boolean logic. Genius!
Finally, no action is associated to the expression, so that the default one (print $0) is applied.
i just want to remove abc 1 and agsi 3 but not iaij... its removing all three..
This complicates things a bit. What is the logic for not removing duplicates of "iaij"? Is it simply an exclusion of some specific strings? Or does it depend on the absence of any other (numeric) field on the same line?