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Cataland: Why the Fuß?

https://doi.org/10.1090/memo/1535

Stump, Christian; Thomas, Hugh; Williams, Nathan (January 2025, 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
Ian G. Macdonald: Works of Art; Hochschild Polytopes; Toric and Permutoric Promotion; Smirnov Words and the Delta Conjectures; The Poincaré-extended ab-Index; Colored Permutation Statistics by Conjugacy Class; Upho Lattices and Their Cores; Pattern Heights and The Minimal Power of q in a Kazhdan-Lusztig Polynomial; Rowmotion Markov Chains; Enumerating The Faces of Split Matroid Polytopes; From Higher Bruhat Orders to Steenrod Cup-i Coproducts; On The Size of Bruhat Intervals; Vines and MAT-Labeled Graphs; Configuration Spaces and Peak Representations; A determinantal Point Process Approach to Scaling and Local Limits of Random Young Tableaux; Generalized Heawood Graphs and Triangulations of Tori; The Somewhere-To-Below Shuffles in The Symmetric Group and Hecke Algebras; Dyck Combinatorics in p-Kazhdan-Lusztig Theory; A Framework Unifying Some Bijections for Graphs and Its Connection to Lawrence Polytopes; Shi Arrangements and Low Elements in Coxeter Groups; Universal Plücker Coordinates for the Wronski Map and Positivity in Real Schubert Calculus; Wilting Theory of Flow Polytopes; Character Factorisations, z-Asymmetric Partitions, and Plethysm; Inhomogeneous Particle Process Defined by Canonical Grothendieck Polynomials; Chow Rings of Matroids as Permutation Representations; Diagram Model for The Okada Algebra and Monoid; Stanley Chromatic Functions and a Conjecture in The Representation Theory of Unipotent Groups; Regular Schur Labeled Skew Shape Posets and Their 0-Hecke Modules; Acyclonestohedra; Determinant of The Distance Matrix of a Tree; Pop-Stack for Cambrian Lattices; The Characteristic Quasi-Polynomials for Exceptional Well-Generated Complex Reflection Groups; Macdonald Characters from a New Formula for Macdonald Polynomials; Antichains in The Representation Theory of Finite Lattices; On The Scaling of Random Tamari Intervals and Schnyder Woods of Random Triangulations (with an Asymptotic D-Finite Trick); Extended Schur Functions and Bases Related by Involutions; Supersolvable Posets and Fiber-Type Arrangements; Levi-Spherical Varieties and Demazure Characters; Non-Symmetric Cauchy Kernels, Demazure Measures, and LPP; Tamari Intervals and Blossoming Trees; Eulerian Polynomials for Digraphs; Framed Polytopes and Higher Categories; Poset Polytopes and Pipe Dreams: Toric Degenerations and Beyond; On The f-Vectors of Poset Associahedra; Rational Catalan Numbers for Complex Reflection Groups; Brewing Fubini-Bruhat Orders; From Geometry to Generating Functions: Rectangulations and Permutations; Growth Diagrams for Schubert RSK; A Signed e-Expansion of The Chromatic Symmetric Function and Some New e-Positive Graphs; All Kronecker Coefficients are Reduced Kronecker Coefficients; Quasipartition and Planar Quasipartition Algebras; Invariant Theory for The Face Algebra of The Braid Arrangement; The Newton Polytope of The Kronecker Product; 0-Whirling and Rowmotion Dynamics on The Chain of V's Poset; A Whitney Polynomial for Hypermaps; The Doubly Asymmetric Simple Exclusion Process, The Colored Boolean Process, and The Restricted Random Growth Model; From Coloured Permutations to Hadamard Products and Zeta Functions; Murnaghan-Type Representations of the Elliptic Hall Algebra; Plane Partitions and Rowmotion on Rectangular and Trapezoidal Posets; Projective Dimension of Weakly Chordal Graphic Arrangements; Realizing the s-Permutahedron via Flow Polytopes; Monomial Expansions for q-Whittaker and Modified Hall-Littlewood Polynomials; Extremal Weight Crystals over Affine Lie Algebras of Infinite Rank; Lascoux Polynomials and Gelfand-Zetlin Patterns; Chain Polynomials of Generalized Paving Matroids; A Galois Structure on The Orbit of Large Steps Walks in The Quadrant; Locally Invariant Vectors in Representations of Symmetric Groups; Excedance quotients, Quasisymmetric Varieties, and Temperley-Lieb algebras; Combinatorial Properties of Triangular Partitions; Asymptotics of Bounded Lecture-Hall Tableaux; Shuffle Theorems and Sandpiles; Real Matroid Schubert Varieties, Zonotopes, and Virtual Weyl Groups; Chromatic Functions, Interval Orders, and Increasing Forests; Crystals for Variations of Decomposition Tableaux; Triangular (q, t)-Schröder Polynomials and Khovanov-Rozansky Homology; qtRSK*: A Probabilistic Dual RSK Correspondence for Macdonald Polynomials; An Extended Generalization of RSK via The Combinatorics of Type A Quiver Representations; Pattern-Avoiding Polytopes and Cambrian Lattices; A Consistent Sandpile Torsor Algorithm for Regular Matroids; Asymptotics of Bivariate Algebraico-Logarithmic Generating Functions; Two-Row Set-Valued Tableaux: Catalan+k Combinatorics; Skein Relations for Punctured Surfaces; Charmed Roots and The Kroweras Complement; Fragmenting any Parallelepiped into a Signed Tiling; Ehrhart Polynomials, Hecke Series, and Affine Buildings; Cluster Algebras and Tilings for The m = 4 Amplituhedron; Quasisymmetric Expansion of Hall-Littlewood Symmetric Functions; Geometry of C-Matrices for Mutation-Infinite Quivers; Higher Specht Polynomials and Tableaux Bijections for Hessenberg Varieties; Charge Formulas for Macdonald Polynomials at t = 0 from Multiline Queues and Diagrams; Slit-Slide-Sew Bijections for Constellations and Quasiconstellations; Crystal Chute Moves on Pipe Dreams; A Regular Unimodular Triangulation of the Matroid Base Polytope; Analogues of Two Classical Pipedream Results on Bumpless Pipedreams; Asymptotic Log-Concavity of Dominant Lower Bruhat Intervals via Brunn-Minkowski Inequality; Combinatorics of Boundary Algebras; The Commutant of Divided Difference Operators, Klyachko's Genus, and the comaj Statistic; Jack Derangements; Framing Lattices and Flow Polytopes; On The Sum of The Entries in a Character Table; Vertex Models for The Product of a Schur and Demazure Polynomial; On The f-Vectors of Flow Polytopes for The Complete Graph; Counting Unicellular Maps under Cyclic Symmetries; Weak Bruhat Interval 0-Hecke Modules in Finite Type; The Monoid Representation of Upho Posets and Total Positivity; Shards for The Affine Symmetric Group; Kromatic Quasisymmetric Functions

Ram, Arun; Macdonald, Ian G; Pilaud, Vincent; Poliakova, Daria; Defant, Colin; Madhukara, Rachana; Thomas, Hugh; Iraci, Alessandro; Nadeau, Philippe; Vanden_Wyngaerd, Anna; et al (September 2024, Séminaire lotharingien de combinatoire)
Gaetz, Christian (Ed.)
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