#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 \([n].\) A composition \(\beta=(\beta_1,\dotsc,\beta_k)\) is a peak composition if all parts except possibly \(\beta_k\) are greater than \(1.\) Under the descent-set bijection between compositions of \(n\) and subsets of \([n-1],\) peak compositions correspond exactly to peak sets. 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].
In degree four, the peak sets are \(\emptyset,\) \(\{2\},\) and \(\{3\}.\) The corresponding peak functions expand in the fundamental quasisymmetric basis as \( \Lambda \) \( \peakQSym_\Lambda(\xvec) \) \( \emptyset \) \( 2\gessel_{1111}+2\gessel_{112}+2\gessel_{121}+2\gessel_{13} +2\gessel_{211}+2\gessel_{22}+2\gessel_{31}+2\gessel_{4} \) \( \{2\} \) \( 4\gessel_{121}+4\gessel_{13}+4\gessel_{211}+4\gessel_{22} \) \( \{3\} \) \( 4\gessel_{112}+4\gessel_{121}+4\gessel_{22}+4\gessel_{31} \)
See also [GZ18].
D. Grinberg and E. A. Vassilieva introduce algebras of extended peaks, building on their work on \(q\)-deformed \(P\)-partitions [GV23]. For each complex root of unity \(q,\) they obtain a subalgebra of quasisymmetric functions with a basis given by a \(q\)-analogue of Gessel’s fundamental quasisymmetric functions. The indexing statistic interpolates between descent sets and peak sets. M. Slattery-Holmes studies pattern-avoiding peak functions [Sla25], answering when the peak-function analogues of pattern quasisymmetric functions indexed by subsets of \(\symS_3\) are symmetric. In the symmetric cases, the paper gives explicit positive expansions in Schur \(Q\)-functions.
Example (Pattern-avoiding peak functions in degree four).
For a set \(\Pi\) of permutation patterns, let \(\mathrm{Av}_n(\Pi)\) be the set of permutations of size \(n\) avoiding all patterns in \(\Pi.\) The pattern-avoiding peak function is \[R_n(\Pi)\coloneqq \sum_{\pi\in\mathrm{Av}_n(\Pi)} \peakQSym_{\Peak(\pi)}.\] Slattery-Holmes classifies the sets \(\Pi\subseteq\symS_3\) for which \(R_n(\Pi)\) is symmetric for all \(n,\) and gives Schur \(Q\)-positive formulas [Thm. 1.1, Sla25]. In degree four, some representative cases are \( \Pi \) \( R_4(\Pi) \) \( \{123\} \) \( 4\schurQ_{(4)} + 5\schurQ_{(3,1)} \) \( \{132,231\} \) \( 8\schurQ_{(4)} \) \( \{123,132,312\} \) \( 2\schurQ_{(4)} + \schurQ_{(3,1)} \) Here \(\schurQ_{(4)}=\peakQSym_{\emptyset}\) and \(\schurQ_{(3,1)}=\peakQSym_{\{2\}}+\peakQSym_{\{3\}},\) so the first line is equivalently \[R_4(\{123\}) =4\peakQSym_{\emptyset} +5\peakQSym_{\{2\}} +5\peakQSym_{\{3\}}.\]
#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 \(w_{i-1} \lt w_i \gt w_{i+1}.\)
Bibliography
- [GZ18]Ira M. Gessel and Yan Zhuang. Shuffle-compatible permutation statistics. Advances in Mathematics, 332:85–141, July 2018.
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@article{GesselZhuang2018, doi = {10.1016/j.aim.2018.05.003}, url2 = {https://doi.org/10.1016/j.aim.2018.05.003}, year = {2018}, month = jul, publisher = {Elsevier {BV}}, volume = {332}, pages = {85--141}, author = {Ira M. Gessel and Yan Zhuang}, title = {Shuffle-compatible permutation statistics}, journal = {Advances in Mathematics} } - [GV23]Darij Grinberg and Ekaterina A. Vassilieva. The algebra of extended peaks. Séminaire Lotharingien de Combinatoire, 89B:Article 46, 2023.
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@article{Li2018, doi = {10.1016/j.jcta.2018.04.003}, url2 = {https://doi.org/10.1016/j.jcta.2018.04.003}, year = {2018}, month = aug, publisher = {Elsevier {BV}}, volume = {158}, pages = {449--491}, author = {Yunnan Li}, title = {On weak peak quasisymmetric functions}, journal = {Journal of Combinatorial Theory, Series A} } - [Sla25]Matthew Slattery-Holmes. Pattern-Avoiding Peak Functions. arXiv:2510.17116, 2025.
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@article{SlatteryHolmes2025x, author = {Matthew Slattery-Holmes}, title = {Pattern-{A}voiding {P}eak {F}unctions}, year = {2025}, eprint = {2510.17116}, url = {https://arxiv.org/abs/2510.17116}, journal = {arXiv e-prints} } - [Ste97]John Stembridge. Enriched $P$-partitions. Transactions of the American Mathematical Society, 349(2):763–788, 1997.
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@article{Stembridge1997, title={Enriched $P$-partitions}, ISSN = {00029947}, author={Stembridge, John}, journal={Transactions of the American Mathematical Society}, url = {http://www.jstor.org/stable/2155395}, volume={349}, number={2}, pages={763--788}, year={1997} }