On the kth smallest part of a partition into distinct parts

A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of n into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of n. In this article, we initiate the study of the kth smallest part of a partition \(\pi \) $\pi$ into distinct parts of any integer n, namely \(s_k(\pi )\) $s_k(\pi )$ . Using \(s_k(\pi )\) $s_k(\pi )$ , we generalize the above result for the kth smallest parts of partitions for any positive integer k and show its connection with divisor functions for general k and derive interesting special cases. We also study weighted partitions involving \(s_k(\pi )\) $s_k(\pi )$ with another parameter z, which helps us obtain several new combinatorial and analytical results. Finally, we prove sum-of-tails identities associated with the weighted partition function involving \(s_k(\pi )\) $s_k(\pi )$ .