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 $\mathcal{P}_n.$
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].
See also [GZ18].
Enriched P-partitions
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 $w_{i-1} \lt w_i \gt w_{i+1}.$
References
- [GZ18] Ira M. Gessel and Yan Zhuang. Shuffle-compatible permutation statistics. Advances in Mathematics, 332:85–141, July 2018.
- [Li18] Yunnan Li. On weak peak quasisymmetric functions. Journal of Combinatorial Theory, Series A, 158:449–491, August 2018.
- [Ste97] John Stembridge. Enriched $p$-partitions. Transactions of the American Mathematical Society, 349(2):763–788, 1997.