On the kth smallest part of a partition into distinct parts

2024/02/19 by Rajat Gupta, Gupta, Rajat, Noah Lebowitz-Lockard +3
Mathematics · #05A19 #11P82 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary 11P81 #Secondary 11P84

paper · pdf · doi:10.48550/arxiv.2402.12549

openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 π into distinct parts of any integer n, namely s_k(π). Using s_k(π), 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(π) 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(π).

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