The Break Buddy Problem

For a lifeguard at a crowded city pool, rotating from station to station, those periodic fifteen-minute breaks between stations are precious commodities. Moreover, since these breaks provide some rare quality time with the other break guards, a lifeguard's question is this: given a certain rotation consisting of stations and breaks, how many of one's breaks are shared with each coworker? Abstract algebra comes to the rescue: we show how the answer, for all coworkers art once, can be packaged in a generating function, computed by an easy calculation in the group ring of the cyclic group.