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Creators/Authors contains: "Williams, Nathan"

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  1. ; ; (, Memoirs of the American Mathematical Society)
    The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras. 
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    Free, publicly-accessible full text available January 1, 2026
  2. ; ; ; (, Proceedings of the London Mathematical Society)
    Abstract We solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects. 
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  3. ; ; ; ; ; (, Journal of the London Mathematical Society)
    Abstract Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under theKreweras complementand nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call theKroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition ofcharmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions. 
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    Free, publicly-accessible full text available November 1, 2025
  4. ; ; (, Combinatorica)
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  5. ; ; (, Transactions of the American Mathematical Society)
    We define an action of words in [ m ] n [m]^n on R m {\mathbb {R}}^m to give a new characterization of rational parking functions—they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani’s zeta map on rational parking functions when m m and n n are coprime [Trans. Amer. Math. Soc. 368 (2016), pp. 8403–8445], and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington’s sweep map on rational Dyck paths (see D. Armstrong, N. A. Loehr, and G. S. Warrington [Adv. Math. 284 (2015), pp. 159–185; E. Gorsky, M. Mazin, and M. Vazirani [Electron. J. Combin. 24 (2017), p. 29; H. Thomas and N. Williams, Selecta Math. (N.S.) 24 (2018), pp. 2003–2034]). 
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  6. ; ; (, Algebraic combinatorics)
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  7. ; ; ; (, The Electronic Journal of Combinatorics)
    The Wiener index of a finite graph $$G$$ is the sum over all pairs $(p,q)$ of vertices of $$G$$ of the distance between $$p$$ and $$q$$. When $$P$$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals. 
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  8. ; ; ; ; (, Algebraic Combinatorics)
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  9. ; (, Journal of the Institute of Mathematics of Jussieu)
    Abstract We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author. 
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  10. ; (, Séminaire lotharingien de combinatoire)
    We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author. 
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    Accepted Manuscript Full Text Available