A combinatorial analysis of the average time for open-address hash coding insertion

In analysing a well-known hash-coding method, Knuth gave an exact expression for the average number of rejections encountered by players of a variant of musical chairs. We study a variant more closely related to musical chairs itself and deduce the same expression by a purely combinatorial approach. In an analysis of the average time to insert an item when using openaddress hash-coding, Knuth [1, p. 528-530] reduced the problem to the following question about musical chairs. Givenm chairs arranged in a circle and numbered clockwise from 0 to m− 1, if in turn each of n people arrives at a randomly selected chair (his initial chair) and walks clockwise until he finds an empty chair (his final chair), what is the average number of rejections (chairs found occupied during the search) per player? For m ≥ n, Knuth showed that the average number of rejections is