The symmetric functions catalog

An overview of symmetric functions and related topics

2021-04-29

Peak quasisymmetric functions

The peak quasisymmetric functions were introduced by J. Stembridge in 1997 [Ste97]. They constitute a basis for a graded subring of the quasisymmetric functions.

Given a composition $\alpha,$ recall how to create a corresponding subset $S_\alpha.$ The following definitions are from [Li18]. A set $\Lambda \subset [2,n-1]$ is a peak set if $j \in \Lambda \implies j-1 \notin \Lambda.$ Given a composition $\alpha,$ we let

$P(\alpha) \coloneqq \{ j \in S_\alpha \cap [2,n-1] | j-1 \notin S_\alpha \}.$

be its peak subset. We let $\mathcal{P}_n$ be the set of all peak subsets of $[n].$ The cardinalities of $\mathcal{P}_n$ for $n=1,2,3,\dotsc$ are $1,1,2,3,5,8,13, \dotsc$ i.e., the Fibonacci numbers. The following Mathematica code produces the set $\mathcal{P}_n.$

Mathematica code

Select[Subsets[Range[2, n - 1]], Length[Intersection[#, # - 1]] == 0 &]

The peak quasisymmetric function $\peakQSym_\Lambda(x),$ $\Lambda\in \mathcal{P}_n$ is defined as

$\peakQSym_{\Lambda}(\xvec) = 2^{|\Lambda|+1} \sum_{\substack{ \alpha \vDash n \\ \Lambda \subseteq S_{\alpha} \triangle (S_{\alpha}+1) }} \gessel_{\alpha}(\xvec).$

Here, $\triangle$ denotes symmetric difference. The definition of $\peakQSym_\Lambda(\xvec)$ is also given in [Prop. 3.5, Ste97].

J. Stembridge introduced the notion of Enriched $P$-partitions, analogous to the classical theory of P-partitions [Ste97]. The idea is to replace the role of descents with peaks. Given a linear extension $w$ of some poset $P,$ $i$ is a peak of $w$ if $w_{i-1} \lt w_i \gt w_{i+1}.$