Involutions on Baxter Objects and q-Gamma Nonnegativity

Abstract

Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with a natural involution. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge's ``$q=-1$ phenomenon''. A polynomial $\sum_{i=0}^{n} a_i t^i$ with symmetric coefficients ($a_{n-i}=a_i$) has a unique expansion $\sum_{k=0}^{\lfloor n/2 \rfloor} \gamma_k t^k(1+t)^{n-2k}$, and is said to be \emph{gamma-nonnegative} if $\gamma_k\geq 0$ for all $k$. We either prove or conjecture a stronger $q$-analogue of this property for several polynomials in two variables $t$,$q$, whose $q=1$ specializations are known to be gamma-nonnegative.