## Combinatorics and Graph Theory

•  Bruce Sagan, MSU
•  Lucas atoms
•  09/25/2019
•  3:00 PM - 3:50 PM
•  C304 Wells Hall

We introduce a powerful algebraic method for proving that Lucas analogues are polynomials with nonnegative coefficients. In particular, we factor a Lucas polynomial as $\{n\}=\prod_{d|n} P_d(s,t)$, where we call the polynomials $P_d(s,t)$ Lucas atoms. This permits us to show that the Lucas analogues of the Fuss-Catalan and Fuss-Narayana numbers for all irreducible Coxeter groups are polynomials in $s,t$. Using gamma expansions, a technique which has recently become popular in combinatorics and geometry, one can show that the Lucas atoms have a close relationship with cyclotomic polynomials $\Phi_d(q)$. Certain results about the $\Phi_d(q)$ can then be lifted to Lucas atoms. In particular, one can prove analogues of theorems of Gauss and Lucas, deduce reduction formulas, and evaluate the $P_d(s,t)$ at various specific values of the variables. This is joint work with Jordan Tirrell based on an idea of Richard Stanley.

## Contact

Department of Mathematics
Michigan State University
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C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science