<   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

Orbifold Markov Numbers

Esther Banaian

It is known that Markov numbers can be viewed as a specialization of the cluster variables in the cluster algebra from a once-punctured torus. We consider "orbifold Markov numbers" which are the result of specializing generalized cluster variables in the generalized cluster algebra (in the sense of Chekhov-Shapiro) from a once-punctured sphere with three orbifold points of order three. It is known by Gyoda that these specializations provide all solutions to a certain generalization of the Markov equation. We provide a direct method for computing orbifold Markov numbers, via snake graphs from orbifolds, and discuss some patterns amongst certain sequences of these numbers. This is based on joint work with Archan Sen.




Shards of Coxeter groups and bricks of preprojective algebras

Will Dana

The representation theory of preprojective algebras for the type ADE Dynkin diagrams is closely intertwined with the combinatorics of the associated Coxeter groups. In particular, the King stability domains of bricks (representations without nontrivial endomorphisms) partition the reflecting hyperplanes of the Coxeter group into cones called shards, which previously arose in work of Nathan Reading on the lattice structure of the weak order. Work in progress of David Speyer and Hugh Thomas constructs "shard modules" for arbitrary preprojective algebras, which likewise correspond to shards of infinite Coxeter groups.


In this poster, we generalize to this infinite setting a couple of results expressing how the relative positions of shards influence the associated shard modules. For each wall of a shard, we obtain a short exact sequence expressing the shard module as a generic extension of simpler shard modules meeting at that wall. In the case that the underlying diagram has a long tail (generalizing the type A and D families), we can use this to construct filtrations of shard modules whose subquotients are determined by the position of the shard in a manner similar to the well-behaved type A situation.




Equidistributions around Special Kinds of Descents and Excedances

Bin Han

We consider a sequence of four variable polynomials by refining Stieltjes's continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections, we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a byproduct, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several companion permutation statistics for two bistatistics in a conjecture of Baril and Kirgizov.




Matrix enumeration from curve singularities

Yifeng Huang

I will present my work about counting mutually annihilating matrices and how it is connected to a nodal singularity. Based on this result, a conjecture is formulated for curve singularities in general. I am leading a project on this and positive evidences of the conjecture have been found on the cusp singularity, and further generalization is promising.




Birational Rowmotion and Chain Shifting in the Rectangle Poset

Joe Johnson

Birational rowmotion is a certain birational map that acts on labelings of the elements of a finite poset. On the product of two chains (called the rectangle poset) Musiker and Roby proved an explicit path formula for all iterated applications of birational rowmotion. We reprove and give a simplified statement of the birational rowmotion formula using the Lindström-Gessel-Viennot lemma. From this formula, we show that sums of weights of certain chains shift in the rectangle when we apply birational rowmotion. This is joint work with Ricky Ini Liu.




Toggling Independent Sets of a Cycle Graph

Michael Joseph

In recent years, actions in toggle groups have been of interest in dynamical algebraic combinatorics, due to various phenomena including periodicity, cyclic sieving, and homomesy. Here, we consider the problem of toggling independent sets in cycle graphs. While the first mystery that caught our interest was the observation that the sum vector in every orbit is periodic with an odd period, our exploration of this dynamical system led us to find an infinite abelian "snake group" that acts simply transitively on the "live entries" of each orbit. This allows us to characterize a number of combinatorial properties of the dynamics by studying the topological covering maps between this torsor and finite quotients. Preliminary work has found other toggle actions where the live entries are a torsor for a group, suggesting that this work is a special case of a more general framework, and posing the question of when this phenomenon arises and why. (This is joint work with Colin Defant, Matthew Macauley, and Alex McDonough.)




Snake Graphs from Punctured Orbifolds

Elizabeth Kelley

We extend the snake graph construction of Musker, Schiffler, and Williams to punctured orbifolds, giving an explicit combinatorial formula for the Laurent expansion of any arc or closed curve. Our construction builds on prior work of Banaian-Kelley and Wilson’s reformulation of gamma-symmetric matchings as good matchings of loop graphs. We also define \(T\)-paths for tagged arcs and make progress towards skein relations for punctured surfaces.




Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables

Anthony Lazzeroni

We introduce a new \(P\) basis for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. quasisymmetric powersum basis. Unlike the quasisymmetric power sums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum \(P\) basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.


We note that our \(P\) basis on QSym agrees with the \(P\)-partition power sums of Aliniaeifard, Wang, and van Willigenburg, and its dual basis in the algebra of noncommutative symmetric functions is the Zassenhaus basis."




Shuffle Lattices and Bubble Lattices

Thomas McConville

C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of these structures. In this article we attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. Intriguingly, most of the combinatorics of the bubble lattice can be encoded by means of two simplicial complexes, the noncrossing matching complex and the noncrossing bipartite complex. We present an intriguing relationship between the \(f\)-vectors of these complexes and relate it to the rank-generating function of the shuffle lattice.




Oriented Matroid Circuit Polytopes: Type A

Jodi McWhirter

Matroids, combinatorial structures that generalize the idea of linear independence in vector spaces, were introduced in the 1930s and give rise to several natural constructions of polytopes. Oriented matroids, similarly, yield many of these same constructions. We examine polytopes that arise from the signed circuits of an oriented matroid. We are able to give bounds on the dimension of a family of these polytopes coming from graphical oriented matroids. Moreover, when we look at the polytope constructed from the cocircuits of the oriented matroid generated by the positive roots of the classical type A root system, we can give an explicit description of the polytope, including its Ehrhart theory.




A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

Khanh Duc Nguyen

The Grothendieck polynomials appearing in the K-theory of Grassmannians are analogs of Schur polynomials. We establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials.




Labeled Chip-firing on Binary Tree

Son Nguyen

We study labeled chip-firing on binary tree and some of its modifications. We prove a sorting property of terminal configurations of the process. We also analyze the move poset and prove that this poset is a modular lattice.




A bijection between K-Kohnert diagrams and reverse set-valued tableaux

Jianping Pan

Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux (RSVT) rule for Lascoux polynomials and reverse semistandard Young tableaux (RSSYT) rule for key polynomials. Besides, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with RSSYT. Ross and Yong introduced K-Kohnert diagrams, which are analogues of Kohnert diagrams. Ross and Yong conjectured a K-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between RSVT and K-Kohnert diagrams. This is joint work with Tianyi Yu.




Towards a Green's Theorem for the Burge Correspondence

Joseph Pappe

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first step towards an analogue of Schensted's and Green's result for the RSK insertion which states that the shape of the tableau can be determined from chains of longest increasing subwords in a word. Furthermore, we give a crystal structure on simple graphs of hook shape. The extremal vectors in this crystal are precisely the simple graphs whose degree sequence are threshold and hook-shaped and have a nice characterization. This is joint work with Digjoy Paul and Anne Schilling.




Inequality of a Class of Schur \(Q\)-Functions

Kyle Salois

The problem of determining when two skew Schur \(Q\)-functions are equal is still largely open. It has been studied in the case of ribbon shapes in 2008 by Barekat and van Willigenburg, and this paper approaches the problem for near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture, with evidence, that all Schur \(Q\)-functions of frayed ribbon shape are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the lattice walks version of the shifted Littlewood-Richardson rule discovered in 2018 by Gillespie, Levinson, and Purbhoo.




Multi-parameter quasisymmetric functions

Leonard Schmitz

Time series analysts make use of quasisymmetric functions as polynomial features which are invariant under dynamic time warping. We extend this notion to multiple parameters and thus provide dynamic warping invariants for tensors of arbitrary shape, including time series, images or videos. We show that multi-parameter quasiysymmetric functions are in a certain sense complete, and provide a quasi-shuffle identity by equipping the underlying tensor algebra with a multidimensional quasi-shuffle of words.




Modules of 0-Hecke-Clifford algebras from diagrams

Dominic Searles

We give a method for constructing modules of 0-Hecke algebras and 0-Hecke-Clifford algebras from fillings of box diagrams. We use this to answer a question of Jing and Li, to define a peak algebra analogue of the quasisymmetric Schur functions, to find a new connection between quasisymmetric Schur functions and Schur Q-functions, and to provide a general framework for known interpretations of families of quasisymmetric functions as characteristics of 0-Hecke modules.




Parking quasi-symmetrizing actions

Adrien Segovia

We define an action of the infinite symmetric group on the set of words of positive integers, called the parking quasi-symmetrizing action, such that the invariants are the elements of the dual of the Hopf algebra \(\mathbf{PQSym}\) of Parking Quasi-Symmetric functions defined by Novelli and Thibon in 2003. Following a work of Hivert, we generalize this action with a parameter \(r\in \mathbb{N}^*\cup \{\infty\}\) and prove that we obtain an infinite chain of nested graded Hopf subalgebras \[\textbf{PQSym}^* = {\textbf{PQSym}^*}^1 \supseteq {\textbf{PQSym}^*}^2 \supseteq ... \supseteq {\textbf{PQSym}^*}^r \supseteq ... \supseteq {\textbf{PQSym}^*}^{\infty} .\]


We give some properties of these new Hopf algebras: basis, Hilbert series, formula for the coproduct and a way to compute the product. The case \(r=\infty\) is particularly interesting since the dimensions of the homogeneous components are related to maximal decreasing subtrees, giving us a new point on view on enumerative results about these sets of trees.




Type B \(q\)-Stirling numbers

Joshua Swanson

The Stirling numbers of the first and second kind are classical objects in enumerative combinatorics which count the number of permutations or set partitions with a given number of cycles or blocks, respectively. Carlitz and Gould introduced \(q\)-analogues of the Stirling numbers of the first and second kinds, which have been further studied by many authors including Gessel, Garsia, Remmel, Wilson, and others, particularly in relation to certain statistics on ordered set partitions. Separately, type B analogues of the Stirling numbers of the first and second kind arise from the study of the intersection lattice of the type B hyperplane arrangement. We combine the two directions and introduce new type B \(q\)-analogues of the Stirling numbers of the first and second kinds. We discuss connections between these new \(q\)-analogues and generating functions identities, inversion and major index-style statistics on type B set partitions, and aspects of super coinvariant algebras which provided the original motivation for the definition. This is joint work with Bruce Sagan.




A symmetric function lift of torus link homology

Andy Wilson

Given any positive integers \(M\) and \(N\), let \(k = \gcd(M,N)\), \(m = M/k\), and \(n=N/k\). We define a symmetric function \(L_{M,N}\) that specializes to the generating function of the triply-graded Khovanov-Rozansky homology of the \(M,N\)-torus link. We give combinatorial and recursive formulations for \(L_{M,N}\) and conjecture that \(L_{M,N}\) is related to \(\mathbf{Q}_{m,n}^{k}(1)\), where \(\mathbf{Q}_{m,n}\) is a symmetric function operator appearing in the elliptic Hall algebra and the Rational Shuffle Theorem.