Order polynomial

Summary

The order polynomial of a finite poset $P$ counts the order-preserving maps from $P$ to a chain with $m$ elements, as a function of $m$. This count is a polynomial in $m$ of degree $\abs{P}$, and its leading coefficient is the number of linear extensions of $P$ divided by $\abs{P}!$. It was introduced by Richard Stanley, who proved a reciprocity theorem linking it to the count of strictly order-preserving maps.