Recent Developments on Spiro's Additive Uniquness Problem

DOI:

Abstract

There are several results showing that certain $q$-series generate various types of partitions or graphical objects. M. Goyal showed that a $q$-series generates split lattice paths, which are graphical representations of split $(n+t)$-color partitions. A. K. Agarwal constructed bijections that map Frobenius symbols to $(n+t)$-color partitions. Motivated by these work, we establish a bijection that maps a partition to a lattice path composed of Plateaus, providing a graphical representation of $(n+1)$-color partition. This bijection can be applied to any partition to transform it into a lattice path. Moreover, we use the bijection in reverse to recover the corresponding partition from a given $(n+1)$-color partition.ed.