Enriched toric $[\vec{D}]$-partitions

TL;DR

This paper introduces enriched toric $[ extbf{D}]$-partitions, extending the theory of enriched $P$-partitions to generate the cyclic peak algebra within cyclic quasi-symmetric functions, highlighting the role of cyclic peak sets.

Contribution

It develops the theory of enriched toric $[ extbf{D}]$-partitions and shows they generate the cyclic peak algebra, a new subring of cyclic quasi-symmetric functions.

Findings

Enriched toric $[ extbf{D}]$-partitions generate the cyclic peak algebra.

The cyclic peak set of permutations is central to the theory.

An associated order polynomial is discussed.

Abstract

This paper develops the theory of enriched toric $[\vec{D}]$-partitions. Whereas Stembridge's enriched $P$-partitions give rises to the peak algebra which is a subring of the ring of quasi-symmetric functions $\text{QSym}$, our enriched toric $[\vec{D}]$-partitions will generate the cyclic peak algebra which is a subring of cyclic quasi-symmetric functions $\text{cQSym}$. In the same manner as the peak set of linear permutations appears when considering enriched $P$-partitions, the cyclic peak set of cyclic permutations plays an important role in our theory. The associated order polynomial is discussed based on this framework.