2019-05-20

## Eulerian quasisymmetric functions

The Eulerain quasisymmetric functions are actually symmetric. They were introduced in [SW10].

### Definition

Let $\bar{1} \lt \bar{2} \lt \dotsb \lt \bar{n} \lt 1 \lt 2 \lt \dotsb \lt n.$ For $\sigma \in \symS_n,$ let $\overline{\sigma}$ be define as the word obtained from $\sigma$ by replacing $\sigma_i$ with $\overline{\sigma}_i$ whenever $\sigma_i \gt i.$ Let

\[ \DEX(\sigma) = \{ i \in [n-1] : \overline{\sigma}(i+1)>\overline{\sigma}(i) \} \]that is, the set of descents in the modified alphabet.

The Eulerian quasisymmetric function indexed by $n$ and $j$ is then described using the Gessel quasisymmetric functions:

\[ \eulerianQ_{\lambda,j}(\xvec) \coloneqq \sum_{\substack{\sigma \in \symS_n \\ \type(\sigma) = \lambda}} \gessel_{n,\DEX(\sigma)}(\xvec). \]**Example.**

We have that

\begin{align*} \eulerianQ_{33,2}(\xvec) &= \gessel_{\emptyset,6} + \gessel_{\{2\},6} + \gessel_{\{3\},6} + \gessel_{\{4\},6} + \gessel_{\{1,3\},6} \\ &+\gessel_{\{1,4\},6} + \gessel_{\{1,5\},6} + \gessel_{\{2,4\},6} + \gessel_{\{2,5\},6} + \gessel_{\{3,5\},6}. \end{align*}### Plethystic formula

**Theorem (See [SW10]).**

Let $\lambda$ have $m_i$ parts of size $i.$ Then

\[ \sum_{j=0}^{|\lambda|-1} \eulerianQ_{\lambda,j}(\xvec)t^j = \prod_{i \geq 1} \completeH_{m_i}\left[ \sum_{j=0}^{i-1} \eulerianQ_{(i),j} t^j \right]. \]### Schur positivity

In [HW12] it is proved that $\eulerianQ_{\lambda,j} = \frobChar V_{\lambda,j},$ that is, it is the Frobenous characteristic of a certain vector space $V_{\lambda,j}$ spanned by forests of marked trees. This implies Schur positivity.

### Power-sum positivity

In [SSW11], it is proved that $\eulerianQ_{(n),j}$ expand positively in the power-sum basis.

**Conjecture (Alexandersson (2018)).**

It seems like all $\eulerianQ_{\lambda,j}$ are $\powerSum_\mu$-positive.

The stronger statement, being positive in the complete homogenous basis, is not true. For example,

\[ \eulerianQ_{(6),3} = \completeH_{321} - \completeH_{411}+2\completeH_{42} + \completeH_{51}. \]### Cyclic sieving phenomena

Let $\symS_{\lambda,j}$ be the subset of permutations in $\symS_n$ with cycle type $\lambda$ and exactly $j$ excedances. In [SSW11], it is shown that

\[ \left(\symS_{\lambda,j}, C_n, \sum_{\sigma \in \symS_{\lambda,j}} q^{\maj(\sigma) - \exc(\sigma)} \right) \]exhibit the cyclic sieving phenomena, where $C_n$ act by conjugation.

**Conjecture (Alexandersson (2018)).**

Let $\lambda$ be a partition of $n.$ Then

\[ \eulerianQ_{\lambda,j}(1,q,q^2,\dotsc,q^{n-1}) \]evaluates to non-negative integers at $q = e^{2 \pi i k/n},$ and there should be some action $C_n$ that completes this to a cyclic sieving phenomena.

## Colored Eulerian quasisymmetric functions

In [Hya12], the notion of Eulerian quasisymmetric functions is generalized to the wreath product of
a cyclic group and the symmetric group, also known as the group of *colored permutations*.
The notion of descents, fixed points and so on generalizes naturally to this setting,
and thus allow for a definition of *colored Eulerian quasisymmetric functions*.

A special case would be to choose the cyclic group to be $\setZ_2,$ which can then be called a type $B$ Eulerian quasisymmetric function.

In this preprint they are shown to be symmetric. These are believed to be symmetric and satsify a cyclic sieving phenomena as above.

## References

- [HW12] Anthony Henderson and Michelle L. Wachs. Unimodality of Eulerian quasisymmetric functions. Journal of Combinatorial Theory, Series A, 119(1):135–145, January 2012.
- [Hya12] Matthew Hyatt. Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups. Advances in Applied Mathematics, 48(3):465–505, March 2012.