Equality of P-partition generating functions

TL;DR

This paper investigates when two labeled posets produce identical quasisymmetric generating functions, providing necessary and sufficient conditions, and classifying all equalities for posets with few linear extensions.

Contribution

It offers a partial classification of equality conditions for P-partition generating functions, advancing understanding in the combinatorics of labeled posets.

Findings

Identified necessary conditions for equality of generating functions.

Established sufficient conditions for equality.

Classified all equalities for posets with few linear extensions.

Abstract

To every labeled poset (P,\omega), one can associate a quasisymmetric generating function for its (P,\omega)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P,\omega) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of linear extensions.