Combinatorics Seminar

Speaker: Semin Yoo

Title: Combinatorics of quadratic spaces over finite fields

Abstract: In this talk, I will talk about the combinatorial structures associated with quadratic spaces over finite fields. I will first introduce a new isometric invariant of combinatorial type in $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$, where $q$ is an odd prime power. Using counts from this invariant, a new binomial coefficient, called the dot-binomial coefficient $\binom{n}{k}_{d}$, will be defined. The dot-binomial coefficient $\binom{n}{k}_{d}$ counts the number of $k$-dimensional quadratic subspaces of Euclidean type in $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$, and behaves like the $q$-binomial coefficient. The similarities and differences between the $q$-binomial coefficient $\binom{n}{k}_{q}$ and the dot-binomial coefficient $\binom{n}{k}_{d}$ will be discussed in this talk. Additionally, I will talk about the relevant combinatorial properties of $\binom{n}{k}_{d}$.