Algebraic and Topological Combinatorics

Special Session at the Fall Eastern Section Meeting
of the American Mathematical Society


Williams College, Williamstown, MA
October 13-14, 2001



Louis J. Billera
(Cornell University)


Quasisymmetric functions and Eulerian enumeration


Abstract [ps] [pdf]: We describe some links between enumeration of chains in Eulerian posets and questions about representations of certain quasisymmetric functions. The commutative {\it peak algebra} $\Pi$ of Stembridge is generated by quasisymmetric functions arising from enriched $P$-partitions. The noncommutative algebra $A_{\mathcal E}$ consists of all chain-enumeration functionals $\sum \alpha_S f_S$ on Eulerian posets. Both have Hilbert series given by the Fibonacci numbers. Bergeron, Mykytiuk, Sottile and van Willigenburg have shown that, with natural coproducts, $\Pi$ and $A_{\mathcal E}$ are dual Hopf algebras. As a consequence, for a rank $n+1$ Eulerian poset $P$, the quasisymmetric function $F(P)=\sum_{S\subset [n]}f_S(P) \thinspace M_S$ is always an element of $\Pi$. We study this pairing explicitly and show that the coefficients of the {\bf cd}-index for Eulerian posets, as elements of $A_{\mathcal E}$, form a dual basis to that given by the weight-enumerators $\Theta_S$ of enriched $P! $-partitions of labelled chains. Thus Eulerian posets $P$ that have nonnegative {\bf cd}-indices are precisely those that are $\Theta$-positive. These include all face posets of convex polytopes and are conjectured to include all Gorenstein* posets.

This is joint work with Samuel K. Hsiao and Stephanie van Willigenburg



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