A Costas array is a permutation array for which the vectors joining pairs of 1s are all distinct. We propose a new 3-D combinatorial object related to Costas arrays: an order n Costas cube is an array (di, j,k) of size n× n× n over ℤ2for which each of the three projections of the array onto two dimensions, namely (Σidi, j,k) and (Σjdi, j,k) and (Σkdi, j,k), is an order n Costas array. We determine all Costas cubes of order at most 29, showing that Costas cubes exist for all these orders except 18 and 19 and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes. We then present constructions for four infinite families of Costas cubes.

We consider problems in the enumeration of sequences suggested by the problem of determining the number of ways of performing a piano composition (Klavierstück XI) by Karlheinz Stockhausen. An explicit formula and a combinatorial proof for the general problem are given.

The author studies permutations of the multiset $\{ 1,1,2,2, \ldots ,m,m,m + 1,m + 2, \ldots ,n \}$ such that $1,2, \ldots ,n$ occurs as a not-necessarily consecutive subsequence. From the theory of symmetric functions, the generating function for the number of these permutations is known [Goulden and Jackson, Combinatorial numeration, John Wiley, New York, 1983, p. 73]. It is used to obtain a recurrence relation and then to give a purely combinatorial proof of the recurrence.