Some new results for Andrews' Kimberling partitions

Abstract

George E. Andrews (2016) introduced the Kimberling index, $K(\pi )$, of a partition $\pi$ of a positive integer $n$, which is defined as $$K(\pi )=(\text{largest part of }\pi )-(\text{least part of }\pi )-(\text{number of parts of }\pi ).$$ Based on Kimberling index, Andrews defined five partition functions, $K_{>}(n),$ $K_{<}(n),$ $K_{\le }(n),$ $K_{=}(n),$ and $K_{\ge }(n)$, called Kimberling partition functions, which count the numbers of partitions of a positive integer $n$ for which the Kimberling index $K(\pi )$ is $>0$, $<0$, $\le 0$, $=0$ and $\ge 0$, respectively. He also gave the generating functions for $K_{\le }(n),$ $K_{<}(n),$ and $K_{>}(n)$ and established some relations connecting Kimberling partitions and other partition functions. Since then, the Kimberling partition functions and their generating functions remained unexplored. In this paper, we derive generating functions for $K_{=}(n),$ and $K_{\ge }(n)$, and establish some congruence relations of the five Kimberling partition functions by using the method of $q$-series identities.

20 pages