Ph.D. (2003) Cornell University
Abstract: We study the algebraic and enumerative combinatorial aspects of Eulerian posets and their quasisymmetric generating functions. These generating functions span the so-called peak algebra Pi, which originated with Stembridge's theory of enriched P-partitions. Remarkably, many constructs in the peak algebra that are natural in the context of enriched P-partitions are also important from the viewpoint of flag enumeration. For example, we show that the fundamental basis of peak functions arising from enriched P-partitions of chains is precisely the basis that is needed to properly encode the cd-index, a common invariant in the study of convex polytopes and Eulerian posets. As another example, the descents-to-peaks map, which relates the ordinary and enriched theories of P-partitions, turns out to be important for computing the flag-enumerative information of an oriented matroid based on that of the underlying matroid.
We introduce a family of Eulerian posets, called posets of signed order ideals. Each of these posets is defined by appropriately "signing" a distributive lattice. We establish that the flag-enumerative relationship between a poset of signed order ideals and its underlying distributive lattice is completely analogous to that between an oriented matroid and its underlying matroid. Furthermore, we show that these posets are EL-shellable, they have non-negative cd-indices, and their quasisymmetric generating functions enumerate the enriched P-partitions of certain labeled posets.
We analyze the (external) Hopf algebra structure on Pi, showing it to be dual to the concatenation Hopf algebra on letters of odd degree. Consequently, Pi is freely generated as a commutative algebra. Upon closer inspection of the generators, we find that Pi is free as a module over the ring of Schur Q-functions. Much of our analysis builds on basic properties of a monomial-like basis for the peak algebra, as well as a recursively defined basis of eigenvectors for the descents-to-peaks map. We give a simple criterion, in terms of this monomial basis, for determining when an element of Pi is symmetric.