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

Inspired by Gansner’s elegant $k$-trace generating function for rectangular plane partitions, we introduce two novel operators, ${\phi }_z$ 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.