AlCoVE 2023 will be held virtually on Zoom on June 26 – 27, 2023 (Monday and Tuesday).
Past conferences: AlCoVE 2020, AlCoVE 2021, and AlCoVE 2022.
Organizers: Laura Colmenarejo, Maria Gillespie, Oliver Pechenik, and Liam Solus
Conference poster: Download here.
AlCoVE brings together researchers interested in algebraic combinatorics from around the world. Each talk will be 30 minutes and between talks, there will be casual social activities for spending time with your friends and making new friends.
Registration and poster session application
To access the Zoom links, you must first register for the conference:
We will be holding virtual poster sessions in Gather on June 26 and June 27. Applications to present a poster have now closed.
List of Confirmed Speakers:
- Michela Ceria, Politecnico di Bari
- Sylvie Corteel, CNRS, Université Paris Cité
- Colin Defant, MIT
- Rafael González D’León, Loyola University Chicago
- Takeshi Ikeda, Waseda University
- Luc Lapointe, Universidad de Talca
- Chiara Meroni, ICERM
- Leonid Monin, Institute of Mathematics, EPFL
- Bruce Sagan, Michigan State University
- Johanna Steinmeyer, Hebrew University of Jerusalem/Københavns Universitet
- Mariel Supina, KTH Royal Institute of Technology
- Ben Young, University of Oregon
Schedule (subject to change, all times EDT):
The password for Zoom and Gather is the same, and will be sent to registered participants.
JUNE 26 (MONDAY):
| 10:00 - 10:30 AM | Welcome | Zoom link | |||
| 10:30 - 11:00 | Sylvie Corteel | Colored vertex models | Zoom link | ||
| 11:00 - 11:30 | Ice breakers | Zoom link | |||
| 11:30 - noon | Johanna Steinmeyer | Combinatorial Hodge theory and IDP lattice polytopes | Zoom link | ||
| noon - 1:30 | Lunch and poster session A | Gather link | |||
| 1:30 - 2:00 | Bruce Sagan | Chromatic symmetric functions and change of basis | Zoom link | ||
| 2:00 - 2:30 | Critter time | Zoom link | |||
| 2:30 - 3:00 | Leonid Monin | Polyhedral models for K-theory | Zoom link | ||
| 3:00 - 4:00 | Puzzles | Zoom link | |||
| 4:00 - 4:30 | Colin Defant | Permutoric promotion: Gliding globs, sliding stones, and colliding coins | Zoom link | ||
| 4:30 - 5:00 | Coffee break | Gather link | |||
| 5:00 - 5:30 | Takeshi Ikeda | Equivariant homology of the affine Grassmannian of the symplectic group | Zoom link | ||
| 5:30 - 6:00 | Happy hour | Gather link |
JUNE 27 (TUESDAY):
| 10:00 - 10:30 AM | Welcome | Gather link | |||
| 10:30 - 11:00 | Rafael González D’León | On Whitney numbers of the first and second kind, or is it the other way around? | Zoom link | ||
| 11:00 - 11:30 | Critter showcase | Zoom link | |||
| 11:30 - noon | Michela Ceria | Alternatives for the \(q\)-matroid axioms of independent spaces, bases, and spanning spaces | Zoom link | ||
| noon - 1:30 PM | Lunch and poster session B | Gather link | |||
| 1:30 - 2:00 | Chiara Meroni | Cutting polytopes | Zoom link | ||
| 2:00 - 2:30 | Virtual expedition: Math in virtual reality | Zoom link | |||
| 2:30 - 3:00 | Mariel Supina | Invitation to equivariant Ehrhart theory | Zoom link | ||
| 3:00 - 4:00 | Puzzles | Zoom link | |||
| 4:00 - 4:30 | Luc Lapointe | Positivity conjectures for m-symmetric Macdonald polynomials | Zoom link | ||
| 4:30 - 5:00 | Coffee break | Gather link | |||
| 5:00 - 5:30 | Ben Young | A Minecraft tour of the tripartite double dimer model | Zoom link | ||
| 5:30 - 6:00 | Happy hour | Gather link |
Posters presentations
POSTER SESSION A, JUNE 26 (MONDAY):
| Esther Banaian | Algebras from orbifolds | poster |
| Aritra Bhattacharya | The Clebsch-Gordan rules for Macdonald polynomials | poster |
| Trung Chau | Barile-Macchia resolutions | poster |
| Patty Commins | Invariant theory for the free left-regular band and a \(q\)-analogue | poster |
| Bishal Deb | Continued fractions using a Laguerre digraph interpretation of the Foata–Zeilberger bijection | poster |
| Danai Deligeorgaki | Inequalities for \(f^*\)-vectors of Lattice Polytopes | poster |
| Yifeng Huang | Spiral shifting operators and punctual Quot schemes on cusp | poster |
| Aryaman Jal | Polyhedral geometry of bisectors and bisection fans | poster |
| V. Sathish Kumar | Decomposition of certain Demazure characters twisted by roots of unity | poster |
| Nishu Kumari | Factorization of classical characters twisted by roots of unity | poster |
| Ezgi Kantarcı Oğuz | Oriented posets | poster |
| Weihong Xu | Quantum K-theory of incidence varieties | poster |
POSTER SESSION B, JUNE 27 (TUESDAY):
Abstracts of talks and posters
Michela Ceria
Alternatives for the \(q\)-matroid axioms of independent spaces, bases, and spanning spaces
\(q\)-matroids generalize matroids substituting "finite sets" with "finite dimensional vector spaces". In the case of \(q\)-matroids, the axioms for independent spaces, spanning spaces and bases are not the straightforward \(q\)-analogue of the classical axioms, because if we take the straightforward \(q\)-analogue of the classical axioms we don't get a \(q\)-matroid. The problem has been solved by adding a fourth axiom, but can we still describe independent spaces, spanning spaces and bases with only three axioms? We will show that the answer to this question is positive and we have indeed two ways to do that. Moreover we see that we can show this way direct cryptomorphisms between independent spaces and circuits and between independent spaces and bases.
Sylvie Corteel
Colored vertex models
I will explain how Macdonald and modified Macdonald polynomials and generalizations can be defined combinatorially using vertex models and how their symmetry is a trivial consequence of the Yang-Baxter equation. This also gives rise to nonsymmetric versions. This is joint work with David Keating (Wisconsin).
Colin Defant
Permutoric promotion: Gliding globs, sliding stones, and colliding coins
We introduce toric promotion as a cyclic variant of Schützenberger's famous promotion operator. Toric promotion acts on the labelings of a graph \(G\); it is defined as the composition of certain toggle operators, listed in a natural cyclic order. We will show that the orbit structure of toric promotion is surprisingly nice when \(G\) is a forest. We then consider more general permutoric promotion operators, which are defined as compositions of the same toggle operators, but in permuted orders. When \(G\) is a path graph, we provide a complete description of the orbit structures of all permutoric promotion operators, showing that they satisfy the cyclic sieving phenomenon. The first half of the proof requires us to introduce and analyze new broken promotion operators, which can be interpreted via globs of liquid gliding on a path graph. For the latter half of our proof, we reformulate the dynamics of permutoric promotion via stones sliding along a cycle graph and coins colliding with each other on a path graph. This is based on joint work with Rachana Madhukara and Hugh Thomas.
Rafael González D’León
On Whitney numbers of the first and second kind, or is it the other way around?
The Whitney numbers of the first and second kind are a pair of poset invariants that are relevant in various areas of mathematics. One of the most interesting appearances of these numbers is as the coefficients of the chromatic polynomial of a graph. They also appear as counting regions in the complement of a real hyperplane arrangement. In this talk, I will share a very curious phenomenon: sometimes the Whitney numbers of the first and second kind of a poset happen to be also the Whitney numbers of the second and first kind but of a different poset. We call this phenomenon Whitney duality and to find examples we rely on the techniques of edge labelings and quotient posets. I will present some key results in the theory of Whitney duality and in particular recent results regarding nonuniqueness. Joint work with Josh Hallam and Yeison Quiceno.
Takeshi Ikeda
Equivariant homology of the affine Grassmannian of the symplectic group
Schubert calculus of the affine Grassmannian \(\mathrm{Gr}_G\) of the symplectic group \(G=\mathrm{Sp}_{2n}(\mathbb{C})\) was studied by Lam, Schilling, and Shimozono. We extend some of their results to torus equivariant setting. The equivariant homology ring is identified with a certain subring of the ring of the dual factorial \(P\)-functions of Nakagawa-Naruse. We introduce equivariantly deformed symmetric functions representing the Schubert classes by using the left divided difference operators. Our construction is based on the universal centralizer of type \(B\), which is also related to the type \(B\)-Toda lattice. Thus, according to the Peterson isomorphism, we have an explicit isomorphism between the equivariant homology of \(\mathrm{Gr}_G\) and the equivariant quantum cohomology ring of the flag variety \(G/B\) of the symplectic group. This is based on joint work with Mark Shimozono, Shinsuke Iwao.
Luc Lapointe
Positivity conjectures for \(m\)-symmetric Macdonald polynomials
We will explain how the construction of Macdonald polynomials in superspace can be naturally extended to define \(m\)-symmetric Macdonald polynomials (which contain as special cases the non-symmetric Macdonald polynomials whose indexing partition is of length smaller than \(m\)). The \(m\)-symmetric Macdonald polynomials have many properties analogous to that of the Macdonald polynomials (scalar product, reproducing kernel, Pieri rules, etc). Most importantly, we conjecture that the \(m\)-symmetric Macdonald polynomials are positive (after a plethystic substitution) when expanded in a certain basis of \(m\)-symmetric Schur functions (now depending on the parameter \(t\)). We will explain how recursion relations among these generalized (\(q,t\))-Kostka coefficients as well as generalizations of Butler's rule seem to present a path to understanding the usual (\(q,t\))-Kostka coefficients.
Chiara Meroni
Cutting polytopes
We obtain a parametric, semialgebraic description of properties of the hyperplane sections of a polytope. Using this structure, we provide algorithms for the optimization of several combinatorial and metric properties over all hyperplane slices of a polytope. This relies on four fundamental hyperplane arrangements. This is a joint work with Marie-Charlotte Brandenburg (Max Planck Institute for Mathematics in the Sciences, Leipzig) and Jesús A. De Loera (University of California, Davis).
Leonid Monin
Polyhedral models for K-theory
Every homogeneous polynomial \(f\) on a vector space \(V\) defines a commutative, graded algebra which satisfies Poincare duality via Macaulay duality. One particularly interesting example of this construction is when \(f\) is the volume polynomial on a suitable space of (virtual) polytopes. In this case the algebra \(A_f\) recovers cohomology rings of toric or flag varieties.
In my talk, I will recall these results and present their recent generalizations. In particular, I will explain how to associate an algebra with Gorenstein duality to any function \(g\) on a lattice \(L\). In the case when \(g\) is the Ehrhart function on a lattice of integer (virtual) polytopes, this construction recovers K-theory of toric and full flag varieties.
Bruce Sagan
Chromatic symmetric functions and change of basis
For a graph \(G\), let \(X(G)\) be Stanley’s chromatic symmetric function. We show that if \(e_\lambda\) appears with nonzero coefficient in the elementary symmetric function expansion for \(X(G)\), then the shape of \(\lambda\) gives bounds on the independence number and clique number of \(G\). This is done by first considering the expansion of \(X(G)\) in terms of monomial symmetric functions and then doing a basis change. This permits us to make progress on the \((3+1)\)- free Conjecture of Stanley and Stembridge as well as give simple proofs of previous results. This is joint work with Foster Tom.
Johanna Steinmeyer
Combinatorial Hodge theory and IDP lattice polytopes
One of the fundamental questions of Ehrhart theory lies in characterizing the possible \(h^*\)-polynomials of lattice polytopes. One discovers especially nice structures if the polytope is reflexive and IDP. In this case, Hibi and Ohsugi conjectured that the coefficients of the \(h^*\)-polynomial are always unimodal. We give a proof of this conjecture by establishing generic anisotropy and strong Lefschetz properties in the associated semigroup algebras.
I will give an overview of the results and techniques. No prior knowledge of combinatorial Hodge theory is assumed. Based on joint work with Adiprasito, Papadakis, and Petrotou.
Mariel Supina
Invitation to equivariant Ehrhart theory
Ehrhart theory is a topic in geometric combinatorics which involves counting the lattice points inside of lattice polytopes. Alan Stapledon (2010) introduced equivariant Ehrhart theory, which combines discrete geometry, combinatorics, and representation theory to give a generalization of Ehrhart theory that accounts for the symmetries of polytopes. In this talk, I will demonstrate an example of equivariant Ehrhart theory and discuss some open problems in the area. I will mention joint work with Federico Ardila and Andrés Vindas-Meléndez (2020) and with Sophia Elia and Donghyun Kim (2022).
Ben Young
A Minecraft tour of the tripartite double dimer model
What's up, mathematicians! Bizarrely, tripartite double dimers appear in two unrelated ways in the enumerative geometry of toric CY threefolds. In recent joint work with Ava Bamforth and Tatyana Benko, we find a simple, purely combinatorial explanation. Both theories involve \(q\)-counting certain \(\mathbb{C}[x,y,z]\)-modules, naturally visualized as piles of little cubes; one applies a "folklore" correspondence to these cubes to view them as dimer configurations. Tripartiteness happens when you "take the quotient by the diagonal". For obvious reasons, Minecraft is the ideal tool for this research project; I will use it to give the entire talk. I will also showcase Dimerpaint, a dimer model tool developed by myself and Leigh Foster.
Esther Banaian
Algebras from orbifolds
We discuss two algebras associated to triangulated unpunctured orbifolds with all orbifold points of order three - a gentle algebra and a generalized cluster algebra, in the sense of Chekhov and Shapiro. To each algebra, we associate a map which can be seen as taking arcs on the orbifold to Laurent polynomials. The first map was defined by Caldero and Chapoton; the second is the snake graph map, defined for surfaces by Musiker, Schiffler, and Williams and for orbifolds by B. and Kelley. We show that the outputs of these two maps agree. This poster is based on joint work with Yadira Valdivieso.
Aritra Bhattacharya
The Clebsch-Gordan rules for Macdonald polynomials
We compute the product of a nonsymmetric Macdonald polynomial \(E_{\mu}\) of type \(GL_2\) with a symmetric Macdonald polynomial \(P_{\lambda}\) of type \(GL_2\) and the product of two symmetric Macdonald polynomials \(P_{\lambda}P_{\nu}\) of type \(GL_2\). We make use of techniques from Hecke algebra. Martha Yip gave alcove walk expansions of the products \(E_\mu P_\nu\) and \(P_\mu P_\nu\). In general, for the product \(E_\ell P_m\) for \(SL_2\), the alcove walk expansion of Yip will be a sum over \(2\cdot 3^{\ell-1}\) alcove walks which simplifies to \(2\ell\) terms. Our result provides an explicit closed formula for each of the \(2\ell\) terms directly.
Trung Chau
Barile-Macchia resolutions
There are many ways to explicitly construct free resolutions of monomial ideals over a polynomial ring, e.g., Taylor resolutions. However, it is hard to obtain MINIMAL free resolutions. Batzies and Welker in 2002 showed that homogeneous acyclic matchings induce free resolutions for monomial ideals, provided an algorithm to produce them, and called them Lyubeznik resolutions. In this joint work with Selvi Kara, we introduce an alternate algorithm to produce homogeneous acyclic matchings, and called the induced resolutions Barile-Macchia. We show that Barile-Macchia resolutions are minimal for large classes of monomial ideals.
Patty Commins
Invariant theory for the free left-regular band and a \(q\)-analogue
We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on \(n\) letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its \(q\)-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras and characterize them using Stirling and \(q\)-Stirling numbers.
We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs.
Bishal Deb
Continued fractions using a Laguerre digraph interpretation of the Foata–Zeilberger bijection
A Laguerre digraph is a directed graph on \(n\) vertices such that each vertex has indegree 0 or 1 and outdegree 0 or 1. Thus, the connected components are either directed cycles or directed paths. They are a combinatorial interpretation of the coefficients of the Laguerre polynomials. A Laguerre digraph with no directed paths and only directed cycles is simply the digraph of a permutation in cycle notation. We will begin by introducing Laguerre digraphs.
We then introduce Flajolet's combinatorial theory of continued fractions and state a continued fraction identity for the series \(\sum_{n=0}^{\infty} n!\) due to Euler (1760). There are several proofs known for this identity; we will focus on a bijective proof due to Foata–Zeilberger (1990). We then reinterpret the Foata–Zeilberger bijection using Laguerre digraphs.
In a recent work (arxiv:2304.14487), I solved conjectured continued fractions due to Randrianarivony–Zeng (1996), Sokal–Zeng (2022), and Deb–Sokal (arxiv:2022) using this reinterpretation. We will state some of these results.
Danai Deligeorgaki
Inequalities for \(f^*\)-vectors of Lattice Polytopes
The Ehrhart polynomial \(L_P(m)\) of a lattice polytope \(P\) counts the number of integer points in the \(n\)-th dilate of \(P\). Ehrhart polynomials of polytopes are often described in terms of the vector of coefficients of \(L_P(m)\) with respect to different binomial bases, under which they have non-negative coefficients. Such vectors give rise to the \(h^*\) and \(f^*\)-vector of \(P\), which coincide with the \(h\) and \(f\) vectors of a regular unimodular triangulation of \(P\), whenever it exists. We will see some inequalities that hold among the coefficients of \(f^*\)-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of \(f\)-vectors of simplicial polytopes; e.g., the first half of the \(f^*\)-coefficients increases and the last quarter decreases. Even though \(f^*\)-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property.
Yifeng Huang
Spiral shifting operators and punctual Quot schemes on cusp
(Joint with Ruofan Jiang) We prove a rationality result on the motive (in the Grothendieck ring of varieties) of the Quot scheme of points on the cusp singularity \(x^2=y^3\), extending a well-known phenomenon for the Hilbert scheme of points on singular curves. Our method is based on stratification given by Gröbner theory. The essential combinatorial ingredient behind the rationality is a family of “spiral shifting” operators on \(\{0,1,2,\ldots\}^d\), originally developed by the authors to study the enumeratives of full-rank sublattices of \(Z^d\). The poster will focus on the combinatorics of these operators, which is of independent interest.
Aryaman Jal
Polyhedral geometry of bisectors and bisection fans
Every symmetric convex body induces a norm on the underlying space. The object of our study is the bisector of two points with respect to this norm. A topological description of bisectors is known in the 2 and 3-dimensional cases and recent work of Criado, Joswig and Santos (2022) expanded this to a fuller characterization of the geometric, combinatorial, and topological properties of the bisector. A key object introduced was the bisection fan of a polytope which they were able to explicitly describe in the case of the tropical norm. We discuss the bisector as a polyhedral complex, introduce the notion of bisection cones, and describe the bisection fan corresponding to other polyhedral norms. This is joint work with Katharina Jochemko.
V. Sathish Kumar
Decomposition of certain Demazure characters twisted by roots of unity.
We generalize a theorem of Littlewood concerning the factorization of Schur polynomials when their variables are twisted by roots of unity. There have been several generalizations of this result in recent times. We prove that certain families of flagged skew Schur polynomials admit a similar factorization when twisted by roots of unity. This family of polynomials includes, as a special case, a certain family of Demazure characters.
Nishu Kumari
Factorization of classical characters twisted by roots of unity
Fix natural numbers \(n \geq 1\) and \(t \geq 2\). In a joint work with A. Ayyer (J. Alg., 2023), we consider the irreducible characters of polynomial representations of the classical groups of types A, B, C, and D, namely the general linear, symplectic, and orthogonal groups, evaluated at elements \(\omega^k x_i\) for \(0 \leq k \leq t-1\) and \(1 \leq i \leq n\), where \(\omega\) is a primitive \(t\)'th root of unity. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call \(z\)-asymmetric partitions, where \(z\) is an integer which depends on the group. Moreover, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. We also give product formulas for general \(z\)-asymmetric partitions and \(z\)-asymmetric \(t\)-cores. Lastly, we show that there are infinitely many \(z\)-asymmetric \(t\)-cores for \(t \geq z+2\).
Ezgi Kantarcı Oğuz
Oriented Posets
We define oriented posets with corresponding rank matrices, where linking two posets by an edge corresponds to matrix multiplication. In particular, linking chains via this method gives us fence posets, and taking traces gives us circular fence posets.
We give several applications, including resolution of a conjecture by Leclere and Morier-Genoud's on matrices for \(q\)-deformed rationals, new identities between circular rank polynomials, and a new matrix formula for calculating \(q\)-deformed Markov numbers.
Weihong Xu
Quantum K-theory of Incidence Varieties
We prove a conjecture of Buch and Mihalcea in the case of the incidence variety \(X=\text{Fl}(1, n-1, n)\) and determine the structure of its (T-equivariant) quantum K ring. In particular, we derive a positive Chevalley formula and a presentation for the equivariant quantum K ring of \(X\), as well as a positive Littlewood–Richardson rule for the non-equivariant quantum K ring of \(X\). Our proof is via the study of rationality properties of certain Gromov–Witten varieties, which are subvarieties of the Kontsevich moduli space of 3-pointed genus 0 stable maps to \(X\). We also conjecture a presentation for the quantum K ring of any flag variety \(\text{Fl}(r_1, ..., r_k, n)\). Part of this poster is based on joint work with W. Gu, L. Mihalcea, E. Sharpe, and H. Zhang.
Tucker Jerome Ervin
New Hereditary and Mutation-Invariant Property Arising from Forks
A hereditary property of quivers is a property preserved by restriction to any full subquiver. Similarly, a mutation-invariant property of quivers is a property preserved by mutation. Using forks, a class of quivers developed by Warkentin, we introduce a new hereditary and mutation-invariant property. We prove that a quiver being mutation-equivalent to a finite number of non-forks — which we define as having a finite forkless part — is this new property, using only elementary methods.
David Garber
A \(q,r\)-analogue of Stirling numbers of type B of the first kind
The (unsigned) Stirling numbers of the first kind \(s(n,k)\) are defined by the following identity:
\(t(t+1)\cdots (t+n-1)=\sum\limits_{k=1}^n s(n,k) t^k.\)
A known combinatorial interpretation for these numbers is given by considering them as the number of permutations of the set \(\{1,2,...,n\}\) having \(k\) cycles.
Bala presented a generalization of the Stirling numbers of the first kind to the framework of Coxeter groups of type B. We denote these numbers by \(s^B(n,k)\). The definition of these numbers is as follows:
\((t+1)(t+3)\cdots (t+(2n-1))=\sum\limits_{k=0}^n s^B(n,k) \cdot t^k.\)
A generalization of Stirling numbers was given by Broder, which is called the \(r\)-Stirling number. Also, some \(q\)-analogues were given by several authors, see e.g. Sagan and Swanson.
In this work, we suggest a \(q,r\)-analogue for the Stirling numbers of the first kind for the Coxeter groups of type B, together with a combinatorial interpretation and some identities.
Hans Höngesberg
Alternating sign matrices with reflective symmetry and plane partitions: \(n+3\) pairs of equivalent statistics
Vertically symmetric alternating sign matrices (VSASMs) of order \(2n+1\) are known to be equinumerous with lozenge tilings of a hexagon with side lengths \(2n+2,2n,2n+2,2n,2n+2,2n\) and a central triangular hole of size 2 that exhibit a cyclical as well as a vertical symmetry, but no bijection between these two classes of objects has been constructed so far. In order to make progress towards finding such a bijection, we generalize this result by introducing certain natural extensions for both objects along with \(n+3\) parameters and show that the multivariate generating functions with respect to these parameters coincide. The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects. The necessity of natural extensions of the original objects may hint at why it is so hard to come up with an explicit bijection for the original objects.
This is joint work with Ilse Fischer.
Joe Johnson
Plane Partitions of Trapezoid Shape
A plane partition (or P-partition) is a nonnegative integer labeling of a poset which weakly increases as you travel up cover relations. In 1983, Proctor showed that the rectangle poset and its associated trapezoid poset have the same number of plane partitions with maximum label \(k\) for each nonnegative integer \(k\). We give a bijective proof of this result. Our bijection arises from a piecewise-linear, volume-preserving, and continuous map between order polytopes, which distinguishes it from the only other known bijection. This poster is based on joint work with Ricky Liu.
Siddheswar Kundu
Saturation for Flagged Skew Littlewood-Richardson Coefficients
We define an extension of Littlewood-Richardson (LR) coefficients, namely flagged skew Littlewood-Richardson coefficients, involving four partitions and a flag which is a weakly increasing finite sequence of non-negative integers. These coefficients subsume several generalizations of LR coefficients (such as Zelevinsky’s extension of LR coefficients). Then we will establish the saturation property of these coefficients, generalizing the saturation theorems of Knutson-Tao and Kushwaha-Raghavan-Viwanath.
Nadia Lafreniere
Homomesies on Permutations Using the FindStat Database
We performed a systematic study of permutation statistics and bijective maps on permutations in which we identified and proved 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. We identified some maps that are likely to exhibit the homomesy characteristics for several statistics and showed that many maps have no interesting homomesic statistics. On top of the many new homomesy results, our research method is especially interesting: we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies and to suggest theorems to prove. This method has the potential to be applied to several other problems.
Alex McDonough
Signed Determinantal Tilings of Euclidean Space
To answer a question about high-dimensional chip-firing, I previously introduced a family of periodic tilings which are obtained from considering a collection of submatrices based on a given matrix. However, this construction requires a specific class of matrices.
For this project, we generalize this construction to all square matrices using signed tiles. In this context, tiles are allowed to overlap, but each point must be covered by exactly one more positive tile than negative tile. This is joint work with Joseph Doolittle.
Duc-Khanh Nguyen
A generalization of the Murnaghan-Nakayama rule for \(K\)-\(k\)-Schur and \(k\)-Schur functions
We introduce a generalization of \(K\)-\(k\)-Schur functions and \(k\)-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of \(K\)-\(k\)-Schur functions and \(k\)-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for \(k\)-Schur functions, and explains it as a degeneration of the rule for \(K\)-\(k\)-Schur functions. In particular, many other special cases promise to be detailed in the future.
Samrith Ram
Subspace profiles of Linear Operators over Finite Fields
Let \(V\) be a vector space of dimension \(n\) over the finite field \(\mathbb{F}_q\) and let \(\Delta\) be a linear operator on \(V\). A subspace \(W\) of \(V\) is said to have \(\Delta\)-profile \(\mu=(\mu_1,\mu_2,\ldots)\) if \(\dim (W+TW+\cdots+T^{j-1}W)=\mu_1+\cdots+\mu_j\) for each integer \(j\geq 1\). Computing the number of subspaces \(\sigma(\mu)\) with a given \(\Delta\)-profile for an arbitrary operator \(\Delta\) is an open problem. In the case where \(\mu\) has two parts, we derive an explicit formula for \(\sigma(\mu)\) for arbitrary \(\Delta\). In the special case when \(\Delta\) is a regular split semisimple operator, we obtain a new proof of the Touchard–Riordan formula for enumerating chord diagrams by their number of crossings.
Considering subspace profiles leads us to a new family of univariate polynomials indexed by integer partitions. At 1, 0, and -1 these polynomials count set partitions with specified block sizes, standard tableaux of specified shape, and standard shifted tableaux of a specified shape, respectively. These polynomials are generated by a new statistic on set partitions as well as a polynomial statistic on standard tableaux. They allow us to express the \(q\)-Stirling numbers of the second kind defined by Carlitz as sums over standard tableaux and as sums over set partitions. For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated with the \(q\)-Hermite orthogonal polynomial sequence.
Andrew Reimer-Berg
A Generalized RSK for Enumerating Linear Series on \(n\)-pointed Curves
We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-\(d\) morphisms from a general genus-\(g\), \(n\)-marked curve \(C\) to \(\mathbb{P}^r\), sending the marked points on \(C\) to specified general points in \(\mathbb{P}^r\), is equal to \((r+1)^g\) for sufficiently large \(d\). This computation may be rephrased as an intersection problem on Grassmannians, which has a combinatorial interpretation in terms of Young tableaux by the Littlewood-Richardson rule. We give a bijection, generalizing the RSK correspondence, between the tableaux in question and the \((r+1)\)-ary sequences of length \(g\), and we explore our bijection’s combinatorial properties.
We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which \(r = 1\) and several marked points map to the same point in \(\mathbb{P}^1\), the number of morphisms is still \(2g\) for sufficiently large \(d\).
Florian Schreier-Aigner
(-1)-Enumerations of arrowed Gelfand-Tsetlin patterns
Arrowed Gelfand-Tsetlin patterns have recently been introduced to study alternating sign matrices. In this talk, we present a \((-1)\)-enumeration of arrowed GT patterns and of a subclass, which can be expressed by simple product formulas. The \((-1)\)-enumeration in the first case is a one-parameter generalization of the sequence \(1, 4, 60, 3328, \ldots\) that appeared in recent work of Di Francesco. We further provide signless interpretations of our results. The proofs are based on a recent Littlewood-type identity by Fischer and are assisted by the implementation of Sister Celine's algorithm as well as creative telescoping by Koutschan, and by Wegschaider and Riese.
This is based on joint work with Ilse Fischer.
Wahiche
Combinatorial interpretations of the Macdonald identities for affine root systems
We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven infinite families in terms of symplectic and special orthogonal Schur functions. From these results, we are able to derive \(q\)-Nekrasov–Okounkov formulas associated with each family. Nevertheless, we only give results for types \(\tilde{C}\) in this poster.