On $P$-partitions Extended by Two-Rowed Plane Partitions

Abstract

Inspired by Gansner's elegant $k$-trace generating function for rectangular plane partitions, we introduce two novel operators, $\varphi_{z}$ and $\psi_{z}$, along with their combinatorial interpretations. Through these operators, we derive a new formula for $P$-partitions of posets extended by two-rowed plane partitions. This formula allows us to compute explicit enumerative generating functions for various classes of $P$-partitions. Our findings encompass skew plane partitions, diamond-related two-rowed plane partitions, an extended $V$-poset, and ladder poset extensions, enriching the theory of $P$-partitions.