Ballot Sequences and a Determinant of Good's

where A = [uVlnxn with aU = 6i,ji + hi_ ,,j, C = I n-‘J, I is the n x n identity matrix and J is the n x n matrix of all 1’s. This determinant arises in the calculation of cumulants of a statistic analogous to Pearson’s chisquared for a multinomial sample (Sibson [5]), and its value has been conjectured by Good [2]. In this paper we use enumerative methods to prove Good’s conjecture. The particular combinatorial structures we use in the enumeration are generalized ballot sequences, which correspond to random walks with two reflecting barriers. The generating function for this set is determined in Section 2. In Section 3, this generating function is combined with the result that II+ BI = 1 + trace(B) if rank(B) = 1, to give Good’s determinant. A linear recurrence for D,(x) is also obtained. It is worthwhile noting at the outset that the above determinant result is closely related to the following form of the Sherman-Morrison [4] “rank one update” formula