## Combinatorics and Graph Theory

•  Bruce Sagan, MSU
•  Combinatorial interpretations of Lucas analogues
•  09/11/2019
•  3:00 PM - 3:50 PM
•  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.

## Contact

Department of Mathematics
Michigan State University