$P$-Partition Generating Function Equivalence of Naturally Labeled Posets

TL;DR

This paper explores conditions under which different naturally labeled posets share the same $P$-partition generating function, revealing structural invariants and methods to construct non-isomorphic posets with identical functions.

Contribution

It establishes necessary conditions for posets to have identical generating functions and introduces a construction method for non-isomorphic posets sharing the same $P$-partition generating function.

Findings

Posets with the same generating function must have identical antichain counts.

They must share the same shape as defined by Greene.

The paper provides a method to construct non-isomorphic posets with identical generating functions.

Abstract

The $P$-partition generating function of a (naturally labeled) poset $P$ is a quasisymmetric function enumerating order-preserving maps from $P$ to $\mathbb{Z}^+$. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the $P$-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating function.