Distinct r-tuples in integer partitions

We define P-r(q) to be the generating functionwhich counts the total number of distinct (sequential) r -tuples in partitions of n and Q(r)(q, u) to be the corresponding bivariate generating function where u tracks the number of distinct r-tuples. These statistics generalise the number of distinct parts in a partition. In the early part of this paper we develop the tools by finding these generating functions for small cases r = 2 and r = 3. Then we use these methods to obtain P-r(q) and Q(r)(q, u) in the case of general r -tuples. These formulae are used to find the average number of distinct r-tuples for fixed r, as n -> infinity. Finally we show that as r -> infinity, q(-r) P-r(q) converges to an explicitly determined power series.