# Poster session

### Instructions

We have 4x4 (42 inches x 42 inches that fits size A0) poster boards where a printed poster can be placed. We will provide pins. Printed slides would also work. Participants can either bring their printed poster (preferred) or print it on campus at W print.

### Posters

*A New modular symmetric function and its applications: Modular s-Stirling numbers**,*Bazeniar Abdelghafour,**Moussa Ahmia**, José L. Ramírez, Diego Villamizar

In this work, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of weighted lattice path and tilings. We also present some new convolutions involving the complete and elementary symmetric functions. Additionally, we introduce different families of set partitions to give combinatorial interpretations for the modular s-Stirling numbers.

*Connective Constant of Honeycomb Lattice**,***Irha Ali**

This poster will look at the proof of the connective constant of the honeycomb lattice by Hugo Duminil-Copin and Stanislav Smirnov. It will then briefly explain why the only known connective constant is for this lattice.

*On the average number of cycles in conjugacy class products**,***Jesse Campion Loth,**Amarpreet Rattan

We study the product of conjugacy classes C_\alpha C_\beta in the symmetric group S_n. The case \alpha = (n) has received much attention, but for general \alpha and \beta it is not as well understood. It follows immediately from recent work of Chmutov and Pittel that if \alpha has all parts of size at least 3 and \beta has all parts of size at least 2, then the product C_\alpha C_\beta is asymptotically uniform over all even or odd permutations. Their method of proof is using Representation Theory and a recent Character bound of Larsen and Shalev. We show that if \alpha and \beta have all parts of size at least 2, then for all n the average number of cycles in the product C_\alpha C_\beta is between H_n-1 and H_n+1. Our method of proof uses elementary probability theory on a simple class of maps.

*Cyclic sieving for block factorizations of the long cycle**,*Justin Bailey and**Theo Douvropoulos**(UMass Amherst)

Hurwitz knew already in 1889 that there are n^{n-2} minimum length factorizations of the long cycle (123..n) of the symmetric group in transpositions; the same number that counts labeled trees on n vertices. If instead of transpositions, we consider cycles of fixed length, then there are n^{c-1} such factorizations with c factors; this is a special case of the Goulden-Jackson cactus formula.

We study a cyclic action on the set of such factorizations and we count its orbits of different sizes. Our answer is in the form of a cyclic sieving phenomenon (CSP), which means that these orbit sizes are given as polynomial evaluations at roots of unity. We will give background on combinatorial factorizations, explain what a CSP is, and briefly discuss some proof techniques: monodromy of polynomials, and the weighted Bezout’s theorem.

Faces of the generalized Pitman-Stanley polytope,

**William Dugan**, Maura Hegarty, Alejandro Morales, and Annie Raymond (UMass Amherst)

In 1999, Pitman and Stanley introduced a family of polytopes whose integer lattice points biject onto the set of plane partitions of a certain shape with entries in $\{ 0 , 1 \}$, giving descriptions for the vertices of these polytopes and a formula for their volume. In addition, the authors suggested a generalization of their construction depending on $m \in {\mathbb N}$ whose integer lattice points biject onto the set of plane partitions of the same shape having entries in $\{ 0 , 1, ... , m \}$. In this poster, we give further details of this \textbf{generalized Pitman-Stanley polytope}, $PS_n^m(\vec{a})$, demonstrating that it can be realized as the flow polytope of a grid graph with some added structure. We then use the theory of flow polytopes to describe the faces of these polytopes and produce a recurrence for their $f$-vectors. Finally, we apply the transfer-matrix method to obtain the generating functions for the number of vertices of $PS_n^m(1,1,1,...,1)$ for the first few values of $n$. This is joint work with Maura Hegarty, Alejandro Morales, and Annie Raymond.

*Counting isospectral magnetic graphs**,***John Stewart Fabila-Carrasco**

We construct isospectral magnetic graphs using spectral bracketing and vertex contraction. The construction is related to an integer's partitions, giving us a counting of the isospectral graphs. The method is purely based on the spectrum of auxiliary graphs with eigenvalues of sufficiently high multiplicity.

*Arithmetical structures on bidents**,*Kassie Archer, Abigail C. Bishop, Alexander Diaz-Lopez, Luis D. Garcia Puente, Darren Glass,**Joel Louwsma**

An arithmetical structure on a finite, connected graph is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices. Associated to each arithmetical structure is a finite abelian group known as its critical group; this generalizes the notion of the sandpile group of a graph. We study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.

*Pattern avoidance and connectivity in chord diagrams**,***Ali Assem Mahmoud**(Perimeter),**Lukas Nabergall**

We consider hereditary classes of chord diagrams (matchings) that satisfy one of several connectivity properties. Such classes are defined by a set of forbidden subdiagrams or patterns, and we focus on forbidding graphically-defined subdiagrams. Building on recent work, we obtain a decomposition enumerating connected diagrams avoiding so-called bottom cycle subdiagrams. An analogous decomposition can also be used to enumerate many classes avoiding a pair of permutation diagrams of size 3, simplifying prior work. We then present a series of related results and conjectures.

*Universal Partial Cycles on De Bruijn Graphs*, D. Fillmore, B. Goeckner, R. Kirsch,**J. Martin**(Iowa State University), D. McGinnis,

A universal partial cycle for A^n is a cyclic sequence that covers each word of length n over the alphabet A exactly once, like a De Bruijn cycle except that we also allow a wildcard symbol ◇ that simultaneously represents every letter of A. It is well known that a De Bruijn cycle for {0,1,...,a-1}^n can be represented as a Hamiltonian cycle on the De Bruijn graph B(a,n) or as an Eulerian tour of B(a,n-1). I give analogous representations of universal partial cycles on De Bruijn graphs. I present three approaches to constructing new universal partial cycles from old ones. Notably, given any universal partial cycle for {0,1,...,a-1}^n, we show how to construct a universal partial cycle for {0,1,...,ak-1}^n for any natural number k, so each example generates an infinite family of universal partial cycles.

*Longest paths related to Steenrod length of real projective spaces**,***Khanh Duc Nguyen**

We answer an open problem asked by Ravi Vakil in Homotopy theory: Find a combinatorial interpretation of a function f(n) that appears in the Steenrod length of real projective n-space RP^n

*Naruse hook formula for linear extensions of mobile posets**,***GaYee Park**

Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. We give a Naruse type hook-length formula to count linear extensions of such posets by proving a major index $q$-analogue. We also give an inversion index $q$-analogue of the Naruse formula for mobile tree posets.

*A new order on integer partitions**,***Étienne Tétreault**

Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions equivalent to the classical Foulkes conjecture. We establish some of the properties of this relation via the construction of explicit inclusion of modules whose characters correspond to the plethysms considered. We also prove some stability properties for the number of irreducible occurring in these modules as m grows.

*A Subdivision Algebra for a Product of Two Simplices Via Flow Polytopes**,***Matias von Bell**

In 2011, Mészáros introduced the subdivision algebra, in which subdivisions of acyclic root polytopes and certain flow polytopes can be encoded via reductions of polynomials. By considering flow polytopes over certain graphs with bidirectional edges and allowing negative flows, we obtain an extension of the subdivision algebra to a class of flow polytopes which are integrally equivalent to a product of two simplices. In particular, for any lattice path $\nu$ from $(0,0)$ to $(a,b)$ using steps of the form $E=(1,0)$ and $N=(0,1)$, we obtain a subdivision of a product of two simplices $\Delta_a \times \Delta_b$, which is dual to a $w$-simplex, where $w$ denotes the number of valleys in the path $E\nu N$. The volume of each cell in this subdivision is given by a $\nu(i)$-Catalan number, where $\nu(i)$ is a cyclic reordering of the steps in $E\nu N$. We then give a polynomial $p_\nu$ in an extended subdivision algebra whose reductions encode refinements of the subdivision induced by $\nu$. As a special case, we exhibit a reduction order for $p_\nu$ giving a triangulation of $\Delta_a \times \Delta_b$ whose dual is the $\nu$-cyclohedron of Ceballos, Padrol, and Sarmiento.