Example:
I would like to say a special thank you to Corbin Collins who kept this project on schedule despite several unexpected delays. You were part coach, part administrator, and part cheerleader; which was just what we needed.
Why are there no articles in front of the nouns?
This expression is analogous to a generic recipe. For example, a pie crust made from "3 parts flour, 2 parts fat, and 1 part liquid". The "<number> part(s)" phrases are determiners -- they take the place of articles.
In the original poster's example, the author does not want to specify the exact mix. (The mix probably varied over time.) Thus, the author omits the numbers, leaving the bare expression "part coach, part administrator, and part cheerleader".
Because part indicates indefiniteness as to how much coach, administrator, cheerleeder that one is.
According to the entries in three dictionaries as below, the word, part. is used as an adverb. ESL learners have to make themselves familiar with its use.
