Linear extensions of ranked posets, enumerated by descents. A problem of Stanley from the 1981 Banff Conference on Ordered Sets

Let P be a naturally labelled, ranked (graded) poset of rank r and cardinality n. Let Hk be the set of linear extensions of P with k descents. An explicit bijection between Hk and Hn−1−r−k is constructed using the involution principle (0⩽k⩽n−1−r). A problem of Richard P. Stanley from 1981 is thereby solved.