Plane Partitions and Their Pedestal Polynomials

For a linear extension $P$ of a partially ordered set $\mathscr S$, we define a multivariate polynomial by counting certain reverse partitions on $\mathscr S$, called $P$-pedestals. We establish a remarkable property of this polynomial: it does not depend on the choice of $P$. For $\mathscr S$ a Young diagram, we show that this polynomial generalizes the hook polynomial.