Lascoux series, parking functions and noncrossing partitions

[2608.15100] Lascoux series, parking functions and noncrossing partitions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.15100 (math) [Submitted on 15 Aug 2026] Title: Lascoux series, parking functions and noncrossing partitions Authors: Alice L. L. Gao , Xin-Bei Liu , Arthur L. B. Yang , James J. Y. Zhao View a PDF of the paper titled Lascoux series, parking functions and noncrossing partitions, by Alice L. L. Gao and 3 other authors View PDF HTML (experimental) Abstract: In the study of the generating series of Demazure characters, Lascoux used isobaric divided differences to define a family of polynomials $\mathcal{E}_{\sigma}(t)$ indexed by permutations $\sigma$, and asked for a satisfactory expression of these polynomials. In this paper we obtain a combinatorial interpretation of $\mathcal{E}_{\sigma}(t)$ for the permutation $\sigma=[2,3,\ldots,n,1]$ or its inverse in terms of the descent statistic of parking functions of length $n-1$. Based on this progress on Lascoux's open problem, we find that the polynomial $\mathcal{E}_{\sigma}(t)$ for this special case coincides with the $h$-polynomial $h(\Delta(\mathrm{NC}_W),t)$ of the order complex of the noncrossing partition lattice associated to the irreducible Coxeter group $W$ of type $A_{n-1}$. We are inspired by this coincidence to give an operator approach to $h(\Delta(\mathrm{NC}_W),t)$ for any finite Coxeter group $W$. As an application, we completely solve an open problem on $h(\Delta(\mathrm{NC}_W),t)$ which was proposed by Athanasiadis, Douvropoulos and Kalampogia-Evangelinou. For any $k$-divisible noncrossing partition poset $\mathrm{NC}^{(k)}_W$, we also obtain the interlacing symmetric decomposition property of the $h$-polynomial $h(\Delta(\mathrm{NC}^{(k)}_W),t)$. Subjects: Combinatorics (math.CO) MSC classes: 05A20, 20F55, 26C10, 52B05 Cite as: arXiv:2608.15100 [math.CO] (or arXiv:2608.15100v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.15100 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xinbei Liu [ view email ] [v1] Sat, 15 Aug 2026 07:51:36 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled Lascoux series, parking functions and noncrossing partitions, by Alice L. L. Gao and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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