Abstracts: Invited Speakers and Poster Presentations

Invited Speakers

Torin Greenwood
North Dakota State University

Asymptotics of Algebraic Generating Functions
Analytic combinatorics provides a pipeline to derive asymptotics for sequences. First, a combinatorial description of a family of objects is translated into a generating function. Then, the form of the generating function determines the sequence’s growth. While the pipeline for univariate and multivariate rational generating functions is well-understood, algebraic generating functions uncover new challenges. We present recent results on algebraic functions, which can be used to enumerate Catalan objects, lattice paths in restricted domains, trees, and probability distributions. We end with an application to RNA secondary structures, where we find that a structural property originally viewed as a biological novelty is actually commonplace for many combinatorial branching structures.

Pamela E. Harris
University of Wisconsin-Milwaukee

Kostant's Partition Function: Support, Structure, and Surprises
Kostant’s partition function is a fundamental object at the crossroads of Lie theory and algebraic combinatorics. Arising in Kostant’s weight multiplicity formula, it counts the number of ways a weight can be expressed as a nonnegative integral combination of positive roots, and thus encodes structure in representations of semisimple Lie algebras. In this talk, we survey a program developing combinatorial and structural perspectives on this function and, in particular, on the support of Kostant’s multiplicity formula. We highlight explicit descriptions in adjoint representations and structural results showing that the support forms an order ideal in the weak Bruhat order. These developments reveal surprising connections to Fibonacci-type phenomena, lattice and polyhedral models, and to multiplex juggling, positioning Kostant’s partition function as a unifying object in algebraic combinatorics.

Reuven Hodges
University of Kansas

How Often Are Two Random Permutations Comparable?
The weak Bruhat order is a natural partial order on permutations, but even very basic global questions about it are surprisingly difficult. In this talk, I will discuss the probability that two independent uniformly random permutations are comparable in weak Bruhat order. We prove that this probability has asymptotic form exp((−1/2 + o(1)) n log n), which significantly improves the previous best upper and lower bounds of Hammett and Pittel. I will explain how the problem can be reduced to counting linear extensions of permutation posets, and how this leads to a partition-based analysis through the Robinson–Schensted–Knuth correspondence and Plancherel measure. The argument also uses the Baik–Deift–Johansson theorem on longest increasing subsequences of random permutations.

Carlos Martínez Mori
University of Colorado-Denver

Zero Dinv Fubini Rankings and Cake-Cutting Geometry
We show that the number of zero dinv Fubini rankings with n competitors and at most l + 1 ties at any rank is given by the maximum number of regions of an l-dimensional cake with n-1 cuts. This is joint work with Pamela E. Harris and Alexander N. Wilson.

George Nasr
Augustana University

How Community Turned Matroids into Polytopes
In this talk, I will survey my work on polytope combinatorics, with a focus on how it interfaced matroids throughout several collaborations. This work started as an exploration of Kazhdan-Lusztig polynomials for sparse paving matroids, with a goal of providing a manifestly positive and combinatorial formula for the coefficients of the polynomial. Over time, this evolved into a study of matroid polytopes and their valuations as a means to expand on the aforementioned work to paving matroids. With this geometric perspective came a rich framework for new mathematical exploration, made possible thanks to the various collaborations and mathematical communities I have interacted with over the last few years. Through this, polytopes (particularly, symmetric polytopes) have become an independent field of my research, without any direct connection to matroids.

Shira Zerbib
Iowa State University

The Chromatic Number s-stable Kneser Graphs
For s ≥ 2, a subset S of [n] is called s-stable if for every i, j in S with i < j, we have min(j-i, n+i-j) ≥ s.  Denote the set of all s-stable k-subsets of [n] by Stab(n,k,s).  A conjecture of Meunier states that for all n ≥ ks, the chromatic number of the Kneser graph KG(Stab(n,k,s)) is n - s(k-1). This conjecture was previously proved by Schrijver for s = 2, by P. Chen for even s ≥ 4, and by J. Jonsson for s ≥ 4 and large enough n. We resolve the conjecture for s = 3 and large enough n. To this end, we prove a stable version of the Hilton-Milner theorem for 3-uniform families. We also provide a new topological proof for the case where s is even. Joint with Wei-Chia Chen and Alex Parker.

Poster Presentations

Saad Awan
University of Kansas

MacMahon Symmetric Functions and Their Computational Framework
MacMahon symmetric functions generalize classical symmetric functions by considering formal power series in several alphabets that are invariant under the diagonal action of the symmetric group. Although classical symmetric functions play a central role in algebraic combinatorics and are well supported in computer algebra systems such as SageMath, MacMahon symmetric functions remain comparatively understudied. We introduce MacMahonSymmetricFunctions, a SageMath package for explicit computation with MacMahon symmetric functions. The package implements several natural bases, basis change algorithms coming from formulas of Doubilet and Rosas, the omega involution, and related structural operations. By making these constructions computationally accessible, the package provides a practical framework for investigating the algebraic and combinatorial properties of MacMahon symmetric functions.

Melissa Beerbower
University of Wisconsin-Milwaukee

Counting L-interval Fubini Rankings Through Parking Outcomes
Fubini rankings with n competitors are n-tuples that encode the conclusion of a race that allows ties. Since Fubini rankings are parking functions, we can study their parking outcomes, which are permutations encoding the final parking order of the cars using the Fubini ranking as a preference list. We establish that the number of Fubini rankings with n competitors having a fixed parking outcome p is given by 2 to the power of (n-k), where k denotes the number of runs in p. We then use this formula to give a new proof for the number of Fubini rankings, which is given by the Fubini numbers. We also consider the set of L-interval Fubini rankings with n competitors, which are Fubini rankings where at most L+1 competitors tie at any rank. We show that the number of L-interval Fubini rankings with n competitors having a fixed parking outcome p is given by a product of a power of two and a product of L-Pingala numbers, where these factors depend only on the lengths of the runs that make up the parking outcome p. We use these results to give a formula for the number of L-interval Fubini rankings with n competitors for all L less than or equal to n.

David Crosby
University of Arkansas

Minimal Algebra Generators of the Symbolic Rees Algebra of Matroidal Ideals
We give a way to rearrange configurations of rubber bands on a graph so that subsets of rubber bands minimaly cover some edges. We give how this generalizes and relates to the symbolic Rees algebra of matroidal ideals.

Spencer Daugherty
University of Colorado Boulder

Shuffle-compatibility for Statistics on Words, Parking Functions, and Set Partitions
Gessel and Zhuang introduced the concept of shuffle-compatibility of statistics on permutations to describe statistics whose multiset of values on the shuffles of two disjoint permutations is determined exactly by the size and statistic values of the two permutations being shuffled. Shuffle-compatibility implies the existence of an algebraic structure on the equivalence classes induced by the statistic. For example, the descent set statistic is shuffle-compatible, and the algebra on the equivalence classes it induces is isomorphic to the Hopf algebra of quasisymmetric functions. We generalize shuffle-compatibility to objects such as words, parking functions, and set partitions using the Hopf monoids associated with these objects. We investigate statistics on these objects, such as inversion set, tie set, and lucky car set. We then define various algebraic structures on the equivalence classes formed by these statistics that are, in many cases, quotients of (or isomorphic to) well-known combinatorial Hopf algebras such as FQSym, WQSym, PQSym, and NCSym.

Lauren Engelthaler
Baylor University

Jeu de Taquin on Infinite Standard Young Tableaux
Young tableaux are fundamental objects in algebraic combinatorics and representation theory, with operations such as promotion and jeu de taquin playing a central role in their structure and applications. While these operations are well understood for finite tableaux, their behavior on infinite tableaux has so far been studied mainly within probabilistic frameworks. We investigate jeu de taquin on infinite standard Young tableaux from a purely combinatorial and dynamical point of view. We analyze the action of jeu de taquin on infinite shapes, describe the structure of inverse images, and classify tableaux exhibiting periodic, pre-periodic, and recurrent behavior. These results extend classical tableau theory to infinite settings and identify connections between combinatorial dynamics and infinite representation-theoretic structures.

Caleb Fernelius
Kansas State University

Generation Functions of W_1+∞ Action on Symmetric Functions
We describe the action of the infinite-dimensional Lie algebra W_1+∞ and its B-type analogue on Schur and Schur Q-functions, respectively, using the framework of formal distributions. We observe an interesting self-duality property possessed by these compact formulas.

Joshua Fullwood
Kansas State University

The Torus Fixed Point Membership Problem for Springer Fibers in Type A
Springer fibers arise as the fibers of a resolution of singularities of the nilpotent cone of a Lie algebra. Their geometry has proven to be fantastically complex and possess rich connections to objects in representation theory and combinatorics. An open question has been the question of when two components parametrized by Young tableau have non-trivial intersection. More broadly, one can ask the question of when a torus-fixed point parametrized by a row standard tableau is in a given irreducible component. These questions have been answered in type A for certain families of Springer fibers, but have remained open in general. We present work in progress toward a decision procedure that works for every component of a Springer fiber having arbitrary Jordan type in the Lie algebra of the special linear group. We sketch some research directions building on current results.

Kimberly Hadaway
Iowa State University

Parking Completions and Volumes of Polytopes
 Parking functions correspond with preferences of n cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. We generalize parking functions to parking completions: Here, we are given that some cars have already parked in a set of spots, which are indexed in a vector t. We then consider a preference vector c, where length of t + length of c = n. If all cars can park, we say that c is a parking completion. Adeniran et al. (2020) state an open problem which connects the number of parking completions to the volumes of Pitman-Stanley polytopes by explicit computation on small values of n. In this talk, we provide a partial solution to this open problem by exploring edge cases.

Tristan Larson
North Dakota State University

Asymptotics of Bivariate Algebraico-logarithmic Generating Functions
We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors.  Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken into indecomposable parts. Asymptotics are quickly computable and can verify combinatorial properties of sequences and assist in randomly generating objects.  While multiple approaches for algebraic asymptotics have recently emerged, we find that the contour manipulation approach can be extended to transcendental generating functions of this form.

Andrés Molina Giraldo
University of Kansas

Finding Real Phase Structures in Tropical Lines
A real phase structure assigns to each facet of a pure, rational polyhedral complex an affine space mod 2 that is parallel to the facet and satisfies an even covering condition. These have been studied for complexes coming from tropical varieties, such as tropical linear spaces. Here we restrict our attention to tropical lines, which can be represented easily by trees. This allows to simplify the problem of finding all possible real structures of a given line to giving all possible drawings of the line inside a disk in such a way there are no crossings.

Caleb Scutt
New Jersey Institute of Technology

Traces and Their Combinatorial Interpretations
A short presentation introducing the concept of Hecke Algebra and symmetric group traces as well as ways of combinatorially interpreting their evaluations through the lens of Young Tableaux.

Varun Shah
University of Washington, Seattle

Flags in Locally Anti-blocking Polytopes
Kalai’s 3^d-conjecture states that a centrally symmetric d-polytope has at least 3^d faces, with equality achieved, for instance, by the cube. A natural strengthening predicts that, more generally, the number of k-chains in the face lattice is also minimised by the cube for all k. Recent works of Sanyal–Winter and Chor establishes the cases of faces and full-flags of this conjecture, respectively, for the class of locally anti-blocking (LAB) polytopes. In this poster, we present ongoing work on studying the combinatorics of LAB polytopes. As an application, we give a new proof of the Sanyal-Winter 3^d lower bound and establish a corresponding result for flags of length two.

Ashley Skalsky
North Dakota State University

Exploring Antisymmetry While Minimizing Monochromatic Arithmetic Progressions
Among all red/blue colorings of {1, 2, . . . , n}, which has the fewest occurrences of three equally spaced integers all colored the same? What do optimal colorings look like as n tends to infinity? Simulating a conjectured asymptotic minimum is easy, but proving it optimal is not. A challenging theoretical roadblock on the integers is ruling out rapidly alternating colorings; however, strict alternations are actually high-performing in other geometries. We develop new analytic methods to handle alternations and use them to analyze colorings of the star. A special property of the conjectured integer coloring is antisymmetry: the left half is colored oppositely to the right. We find that in other geometries, antisymmetry could have multiple meanings, as explored through a series of comprehensive simulations. We also use these simulations to gain a holistic picture of the space of star colorings, including a categorization of locally optimal colorings along with comparisons of their performances.

Camilo Villamil Chalarca
Oklahoma State University

Length Distributions in Coxeter Foldings
When the simple reflections of a Coxeter group are grouped according to a natural admissibility condition, introduced by B. Mühlherr, the resulting subgroup is again a Coxeter system that sits naturally inside the original group. This construction is known as folding, and the subgroup is called a folding subgroup. We investigate how folding subgroups embed into their ambient Coxeter groups and how their elements are distributed by length with respect to the ambient length function. To measure this distribution, we compute the associated length generating function and show that it admits an explicit expression in terms of q-integers.

Han Yin
University of Kansas

The Record Statistic and Forward Stability of Schubert Products
We initiate a probabilistic study of forward stability for products of Schubert polynomials through the record statistic (left-to-right maxima) of permutations. Building on the explicit record formula for forward stability obtained by Hardt and Wallach, we study random pairs of permutations drawn from three natural families: uniform permutations, Grassmannian permutations, and Boolean permutations. For each family, we determine record probabilities and use them to analyze the asymptotic behavior of forward stability. For uniform and Grassmannian permutations, we obtain asymptotics for the mean together with limiting distribution results. For Boolean permutations, we prove linear-order growth of the mean, and our analysis also produces an explicit time-inhomogeneous Markov chain that yields an exact linear-time uniform sampler. Beyond these cases, we prove that the record-set statistic is equidistributed on the avoidance classes of 132 and 231, and consequently the corresponding forward stability distributions coincide. We conclude with conjectures for numerous further permutation classes and a conjectural recursive criterion for when two avoidance classes have the same record-set distribution.