Counting flags in Eulerian posets: The cd-index

The face lattices of convex polytopes and intervals in the Bruhat order of Coxeter groups are examples of Eulerian posets. The flag vector of an Eulerian poset counts the sequences of elements of specified ranks. The
linear relations that hold for the flag vectors of all Eulerian posets are known, and it turns out the dimension of the flag vectors for posets of fixed rank is a number in the Fibonacci sequence. Jonathan Fine discovered that the flag vectors could be efficiently coded in a vector called the \$cd\$-index. This talk discusses the history of the \$cd\$-index, with a focus on inequalities, connections with other combinatorial parameters and relations to algebraic structures.