Involutions on Baxter Objects and q-Gamma Nonnegativity

Thumbnail Image

Persistent link to this item

View Statistics

Journal Title

Journal ISSN

Volume Title


Involutions on Baxter Objects and q-Gamma Nonnegativity

Published Date




Thesis or Dissertation


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.


University of Minnesota Ph.D. dissertation. 2015. Major: Mathematics. Advisor: Vic Reiner. 1 computer file (PDF); 76 pages.

Related to




Series/Report Number

Funding information

Isbn identifier

Doi identifier

Previously Published Citation

Suggested citation

Dilks, Kevin. (2015). Involutions on Baxter Objects and q-Gamma Nonnegativity. Retrieved from the University Digital Conservancy,

Content distributed via the University Digital Conservancy may be subject to additional license and use restrictions applied by the depositor. By using these files, users agree to the Terms of Use. Materials in the UDC may contain content that is disturbing and/or harmful. For more information, please see our statement on harmful content in digital repositories.