Parity Theorems for Statistics on Domino Arrangements

Abstract

We study special values of Carlitz's $q$-Fibonacci and $q$-Lucas polynomials $F_n(q,t)$ and $L_n(q,t)$. Brief algebraic and detailed combinatorial treatments are presented, the latter based on the fact that these polynomials are bivariate generating functions for a pair of statistics defined, respectively, on linear and circular domino arrangements.