## KU Combinatorics Seminar Fall 2022

Friday 8/26
Organizational meeting

Friday 9/2
Vance Gaffar (Baker University)
Counting embeddings of graphs in state space

Abstract: We will define a labeled digraph $$G(b)$$ on infinite binary strings with $$b$$ 1's called $$b$$-ball state space. For a fixed digraph $$g$$ we will calculate the number of embeddings of $$g$$ into $$G(b)$$ as $$b$$ approaches infinity. It is commonly believed that the number of embeddings of $$g$$ into $$G(b)$$ is eventually a polynomial in $$b$$, we will prove this for certain classes of graphs.

Friday 9/9
Hailong Dao
Fractals and syzygies

Abstract: The Sierpinski gasket is a construction that appears in a number of mathematical contexts: fractal geometry, number theory (odd entries in the Pascal triangle), game theory (the Tower of Hanoi), and more. In this talk I will explain a new sighting of this object: when one tries to construct optimal monomial ideals with linear presentations. This is joint work with David Eisenbud.

Friday 9/16
No seminar

Friday 9/23
Aaron Ortiz
The Defective Parking Space: Tableaux and Catalan Convolutions

Abstract: Parking functions and their enumerative properties have been studied in depth in the field of combinatorics and in computer science as they are connected to the study of hashing functions. A well-studied result is the Catalan parameterization of the number of orbits of the standard parking space, the set of parking functions under the action of the symmetric group. We will introduce $$d$$-defective parking spaces, showing that defect is an invariant with regards to permutations. We establish a connection between the orbits of $$d$$-defective parking space and standard Young tableaux of shape $$(n+d, n-d-1)$$. Additionally, we provide a recursive formula for the number of orbits decomposing the $$d$$-defective parking space and establish an interesting link to Catalan convolutions.

Friday 9/30
Brandon Lee
Laplacian Eigenvalues of Bipartite Kneser-Like Graphs

Abstract: The bipartite Kneser-like graph $$G(a,b)$$ is constructed as follows. For any integers $$a>b>1$$, let $$n=a+b+1$$, then let $$\mathcal{A}$$ be a set of all $$a$$-sized subsets of $$[n]$$. Similarly let $$\mathcal{B}$$ be the set of all $$b$$-sized subsets of $$[n]$$. Draw an edge between a vertex in $$\mathcal{A}$$ and a vertex in $$\mathcal{B}$$ if their intersection is disjoint. We conjecture that all the eigenvalues of the Laplacian matrix of this graph, denoted $$L(G(a,b))$$, are all non-negative integers.

We further conjecture a more precise formula for the multiplicity of any eigenvalue and what all the general eigenvalues are. Thus far, we have proved that: every eigenvector has a related pair with a similar eigenvalue and eigenvector, which implies the sequence of multiplicities is symmetric; that all the eigenvalues of $$L(G(a,2))$$ are non-negative integers; established a transformation to a smaller matrix; and showed found a general formation of the eigenvector for any eigenvalue of $$L(G(a,b))$$.

Friday 10/7
No seminar (Fall Break)

Friday 10/14
Marge Bayer
Title TBA

Friday 10/21
TBA

Friday 10/28, Snow 306
Jeremy Martin
Title TBA (perhaps the braid arrangement?)

Friday 11/4
Mark Denker
Title TBA (perhaps Orlik-Solomon algebras or cut complexes?)

Friday 11/11
Tom Mahoney (Emporia State University)
Title TBA

Friday 11/18
Dania Morales
Title TBA

Friday 11/25
No seminar (Thanksgiving)

Friday 12/2
Jin-Cheng Guu (Stony Brook University)
Title TBA (Kazhdan-Lusztig polynomials?)

Friday 12/9
No seminar (Stop Day)

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Mon 9/26/22