Department of Mathematics

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
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824

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

College of Natural Science