• Speaker: Bruce Sagan, MSU
• Title: Combinatorial interpretations of Lucas analogues
• Date: 09/11/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

The Lucas sequence is a sequence of polynomials in $s,t$ defined recursively by $\{0\}=0$, $\{1\}=1$, and $\{n\}=s\{n-1\}+t\{n-2\}$ for $n\ge2$. On specialization of $s$ and $t$ one can recover the Fibonacci numbers, the nonnegative integers, and the $q$-integers $[n]_q$. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of $n$ in the expression with $\{n\}$. It is then natural to ask if the resulting rational function is actually a polynomial in $s$ and $t$ and, if so, what it counts. Using lattice paths, we give combinatorial models for Lucas analogues of binomial coefficients. We also consider Catalan numbers and their relatives, such as those for finite Coxeter groups. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek.

• Speaker: Aklilu Zeleke, MSU
• Title: New and Old Combinatorial Identities Part I
• Date: 10/16/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

Binomial coefficients $n \choose k$ appear in different areas of mathematics (in Pascal's triangle, counting problems and computing probabilities to name few). There are also many identities that involve binomial coefficients. In this talk we will discuss new and old identities that represent positive integers and in some cases real numbers. These identities are derived from studying the asymptotic behavior of the roots of a generalized Fibonacci polynomial sequence given by $F_{j}(x)=x^{j}-...-x-1$.

• Speaker: Robert Bell, MSU
• Title: Variations of cops and robbers on infinite graphs
• Date: 11/06/2019
• Time: 3:00 PM - 3:50 PM
• Place: C304 Wells Hall

The game of cops and robbers is a two player pursuit and evasion game played on a discrete graph G. We study a variation of the classical rules which leads to a different invariant when G is an infinite graph. In this variation, called "weak cops and robbers," the cops win by preventing the robber from visiting any vertex infinitely often. In the classical game, if G is connected and planar, then the cops can always win if there are at least three cops. We prove that this is true in the weak game if G is a locally finite plane graph with no vertex accumulation points.