Fall 2022

- The Combinatorics Seminar meets on
**Fridays, 3-4pm**, in**Snow 302**~~Snow 302~~. (We may sometimes use Zoom instead.) - Organizers: Mark Denker and Jeremy Martin.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**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