Lecture hall partitions and the affine hyperoctahedral group

TL;DR

This paper explores the relationship between lecture hall partitions and the affine hyperoctahedral group, providing a new perspective that translates combinatorial results between these two mathematical structures.

Contribution

It introduces a novel correspondence between lecture hall partitions and elements of the affine hyperoctahedral group, enabling the transfer of results and generating functions.

Findings

Established a new correspondence between lecture hall partitions and affine hyperoctahedral group elements.

Translated generating function formulas into new insights about the affine hyperoctahedral group.

Connected combinatorial statistics across the two domains.

Abstract

In 1997 Bousquet-M\'elou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$. We provide a new view of their correspondence that allows results in one domain to be translated into the other. We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$.