Wis, your point kind of makes sense to me, and at the same time is incorrect since 3415 is simply invalid, no matter what.

This is all theoretical logic, logics in notation. The real world application does not matter, it's not about what might "feel natural". Personally, doing diabolo for me has very little room for feelings, but that is another topic that I am interested in, not discussed here.

My 2 cents is about it being "natural" to write 51 instead of 15 is that fact that no matter how many repetitions the pattern has, it always must start with a 5 if I'm correct. 4 [51] 41 is no different as 4515151515141 except the amount of cycles is not specified, and the 51 is in between brackets to indicate it's exited state, a term in siteswap that basically means any pattern that needs some transitional throws from a normal state pattern to an exited state. But this does not explain the order, except it seems very logical in my eyes. Anyone?