<   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

AlCoVE 2021 will be held virtually on Zoom on June 14 – 15, 2021 (Monday and Tuesday).

Information about last year's AlCoVE is available here.

Organizers: Laura Colmenarejo, Maria Gillespie, Oliver Pechenik, and Liam Solus

AlCoVE aims to bring 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.

To access the Zoom links, you must first register for the conference:

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Poster Session

We will be holding a virtual poster session in Gather on June 15. Poster titles and abstracts are available here.

Schedule (subject to change, all times EDT):

The password for Zoom and Gather is the same, and was sent to you if you registered. Hint: How does Catalan count to 5? (only numbers)

JUNE 14 (MONDAY):

JUNE 15 (TUESDAY):

Abstracts of talks

Uniform triangulations

An intriguing question about face vectors of simplicial complexes asks to determine when the face enumerating polynomial of a triangulation of a simplicial complex $$\Delta$$ has only real roots. A triangulation of $$\Delta$$ is said to be uniform if the face vector of its restriction to a face of $$\Delta$$ depends only on the dimension of that face. This talk aims to briefly discuss the enumerative theory of uniform triagulations and its applications to real-rootedness questions in f-vector theory and Ehrhart theory, as well as to questions about symmetric decompositions. Our results unify and generalize several results in the literature about special classes of triangulations, such as barycentric, edgewise and interval subdivisions.

Pavel Galashin

Positroid Catalan numbers

To each permutation $$f \in S_n$$ one can associate a polynomial $$C_f(q,t)$$ which encodes the bigraded dimensions of the cohomology of the associated open positroid variety. For the top-dimensional positroid variety, we show that the polynomials $$C_f(q,t)$$ coincide with the rational $$q,t$$-Catalan numbers by relating both sides to knot homology. For an arbitrary permutation $$f$$, $$C_f(q,t)$$ is $$q,t$$-symmetric and unimodal. We introduce a class of repetition-free permutations and show that the corresponding specialization $$C_f(1,1)$$ counts Dyck paths avoiding a convex subset of the rectangle. We also show that any convex subset appears in this way. Joint work with Thomas Lam.

Jim Haglund

Schedules and the Delta Conjecture

The Delta Conjecture says that three things are equal; one is a certain symmetric function defined using Macdonald polynomial operators, and the other two are defined combinatorially in terms of weighted parking functions. The two different combinatorial sides are called the rise version and the valley version. Two different proofs have recently appeared that the rise version equals the symmetric function side; it is still an open question whether they are also equal to the valley version. After overviewing some of this, we will show how the valley version has a "Schedule Formula", which expresses the sum over parking functions in a more compact way involving ordered set partitions. Connections with the study of bases for the diagonal and super diagonal coinvariant rings are discussed. This is joint work with Emily Sergel.

Helen Jenne

The combinatorial Pandharipande-Thomas/Donaldson-Thomas correspondence

In this talk I will discuss joint work with Ben Young and Gautam Webb which resolves an open conjecture from enumerative geometry. The conjecture states that two generating functions for plane partition-like objects (the “box-counting” formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon’s generating function for plane partitions. Our proof is combinatorial: it uses the dimer model, the tripartite double-dimer model of Kenyon and Wilson, and the fact that both satisfy a recurrence related to the Desnanot-Jacobi identity from linear algebra. No prior knowledge of enumerative geometry (or the dimer or double-dimer models) is required to understand the talk.

Amy Pang

Monomial bases for combinatorial Hopf algebras

The Hopf algebra of quasisymmetric functions admits a fundamental (F) basis and a monomial (M) basis, related by the refinement poset: $$F_{\alpha} = \sum_{\beta \geq \alpha} M_{\beta}$$. The natural basis for many other Hopf algebras has properties similar to the F basis, e.g. a product described by some notion of shuffle, and a coproduct following some notion of deconcatenation. In this talk, I give axioms for how these generalised shuffles and deconcatentations should interact with the underlying poset so that a monomial-like basis can be analogously constructed, by generalising the approach of Aguiar and Sottile. This monomial basis has a positive product, a cofree coproduct, and is useful for constructing Hopf morphisms. I will demonstrate the construction on a new Hopf algebra of parking functions, realised as labelled binary trees. This is based on "Hopf algebras of parking functions and decorated planar trees", a joint work with Nantel Bergeron, Rafael Gonzalez D'Leon, Shu Xiao Li and Yannic Vargas.

Digjoy Paul

A multiset generalization of (set) partition algebra

In this talk, we introduce a new diagram algebra, called the multiset partition algebra (MPA), based on specific multiset partitions of a multiset. The multiplication rule is governed by the concatenation of weighted bipartite multigraphs. This is a natural generalization of the partition algebra (PA) introduced by Jones and Martin. We obtain a Schur-Weyl duality between MPA and symmetric group acting on certain symmetric tensors. We associate multiset partitions with certain classes of set partitions, which results in an embedding of MPA into PA, hence proof of semisimplicity and cellularity. The dimensions of irreducible modules of MPA are counted by certain semistandard multiset tableaux. We provided a generating function for multiset tableaux in terms of plethysm of Schur functions and hence a recipe to understand the restriction problem. This is joint work with Sridhar Narayanan and Shraddha Srivastava.

Emmanuella Sandratra Rambeloson

Schur-positivity of some trees

We show that there is no tree $$T$$ on 20 vertices such that $$T$$ has a vertex of degree 10 and the chromatic symmetric function of $$T$$ is Schur-positive. This provides a counterexample to a conjecture of Dahlberg, She and van Willigenburg.

Einar Steingrímsson

Permutations, moments, measures

Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, as a continued fraction, for a 14-parameter family of integer sequences and interpret these in terms of combinatorial statistics on permutations and other combinatorial objects. Special cases include several classical and noncommutative probability laws, and a substantial subset of the orthogonalizing measures in the q-Askey scheme. This captures a variety of combinatorial sequences and characterizes the moment sequences associated to distributions of the numbers of occurrences of (classical and vincular) permutation patterns of length three. This connection between pattern avoidance and classical and noncommutative probability is among several consequences that generalize and unify previous results in the literature. The fourteen combinatorial statistics further generalize to colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points, introduced here. This is joint work with Natasha Blitvić.

Akiyoshi Tsuchiya

Canonical triangulations of enriched order polytopes

Stanley introduced a lattice polytope arising from a finite partially ordered set, which is called an order polytope, and constructed a canonical triangulation of an order polytope. On the other hand, Ohsugi and I introduced an enriched version of order polytopes, which is called an enriched order polytope, from a viewpoint of the theory of enriched P-partitions. In this talk, I construct an explicitly unimodular triangulation of an enriched order polytope as an analogy of the Stanley's canonical triangulation. In particular, I show that our triangulation has a $$\gamma$$-positive $$h$$-polynomial.

This talk is based on joint work with Soichi Okada.

Stephanie van Willigenburg

Schur functions in noncommuting variables

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions, in the algebra of symmetric functions, Sym. In this talk we answer this by introducing Schur functions in noncommuting variables, which naturally refine Rosas-Sagan Schur functions, in addition to having many analogous classical properties that we will discuss.

This is joint work with Farid Aliniaeifard and Shu Xiao Li and requires no prior knowledge.

Cynthia Vinzant

Matroids, log-concavity, and expanders

Matroids are combinatorial objects designed to capture independence relations on collections of objects, such as linear independence of vectors in a vector space or cyclic independence of edges in a graph. Recent work by several independent authors shows that the multivariate basis-generating polynomial of a matroid is log-concave as a function on the positive orthant. In this talk, I will describe some of the underlying combinatorial and geometric structure of such log-concave polynomials and a connection with random walks on the faces of certain simplicial complexes, including the independence complex of a matroid. One consequence of this theory is a proof of the Mihail-Vazirani conjecture that the basis exchange graph of any matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

Lauren Williams

The TASEP on a ring, Schubert polynomials, and evil-avoiding permutations

The inhomogeneous TASEP on a ring is a Markov chain of weighted particles hopping on a ring, in which the probability that two particles interchange depends on the weight of those particles. If each particle has a distinct weight, then we can think of this as a Markov chain on permutations. In this case it turns out that the steady state probability of each evil-avoiding permutation $$w$$ is proportional to a product of $$k$$ Schubert polynomials, where $$k$$ is the number of descents of $$w^{-1}$$. Conjecturally other probabilities are also connected to Schubert polynomials. This is based on joint work with Donghyun Kim.