Div Curl Grad and Fibonacci

In the final exam in problem 3, we looked at all 27 cases to combine the differential operators div curl and grad. We have had quite a few discussions internally whether this is an appropriate problem. Grinding through many cases is not considered enlightening in general. But there are reasons. The case leads to an interesting combinatorial problem:
In how many meaningful way can one build a sentence of length k with the words div, curl, grad?


Here is the answer in a tweet on TikTok (a platform that is obviously is not yet very math heavy). Here is something written down [PDF] and here is something on a chalk board: