Reciprocity between partitions and compositions

TL;DR

This paper explores the reciprocal relationships between partitions and compositions with bounded gaps, extending previous work by analyzing their generating functions and associated matrices.

Contribution

It introduces new reciprocal relations for partitions and compositions with bounded gaps, expanding the theoretical understanding of their generating functions and matrix representations.

Findings

Generated functions satisfy reciprocal relations

Matrices associated with partitions and compositions exhibit reciprocity

Extends previous work on bounded gap structures

Abstract

In this paper, we extend the work of Andrews, Beck and Hopkins by considering partitions and compositions with bounded gaps between each pair of consecutive parts. We show that both their generating functions and two matrices determined by them satisfy certain reciprocal relations.