of the American Mathematical Society

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

Related material:

- Louis J. Billera, Samuel K. Hsia, Stephanie van Willigenburg

Peak quasisymmetric functions and Eulerian enumeration

Back to the mainpage.

last updated: August 16, 2001