Distinct r-tuples in integer partitions

Abstract

We define $P_{r}(q)$ to be the generating function which 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 $n\rightarrow \infty$. Finally we show that as $r\rightarrow \infty$, $q^{-r}P_{r}(q)$ converges to an explicitly determined power series.