My primary research interests are in the interplay between combinatorics and algebraic structures. More specifically, I study the combinatorics of Coxeter groups and their associated Hecke algebras, Kazhdan-Lusztig theory, generalized Temperley-Lieb algebras, diagram algebras, and heaps of pieces. By employing combinatorial tools such as diagram algebras and heaps of pieces, one can gain insight into algebraic structures associated to Coxeter groups, and, conversely, the corresponding structure theory can often lead to surprising combinatorial results. More recently, my research has expanded into combinatorial game theory.

The combinatorial nature of my research naturally lends itself to collaborations with undergraduate students, and my goal is to incorporate undergraduates in my research as much as possible.

My interests also include the scholarship of teaching and learning (SoTL) with a focus on inquiry-based learning (IBL) as an approach to teaching/exploring mathematics. I am currently a Special Projects Coordinator for the Academy of Inquiry-Based Learning and a mentor for several new IBL practitioners. Moreover, I actively give talks and organize workshops on the benefits of IBL as well as the nuts and bolts of how to implement this approach in the mathematics classroom.

You can find a recent version of my curriculum vitae here.

- G. Cerbai, A. Claesson, D.C. Ernst, and H. Golab. Pattern-avoiding Cayley permutations via combinatorial species, Part 2.
- B.J. Benesh, D.C. Ernst, M. Meyer, S.K. Salmon, and N. Sieben. Impartial geodetic destroying games on graphs.
- J. Barnes, J. Breland, D.C. Ernst, R. Perry. Braid graphs in simply-laced triangle-free Coxeter systems are median.
- F. Awik, F. Burkhart, H. Denoncourt, D.C. Ernst, T. Rosenberg, and A. Stewart. Enumerating signed permutations by reversal distance.

- G. Cerbai, A. Claesson, D.C. Ernst, and H. Golab. Pattern-avoiding Cayley permutations via combinatorial species. [arXiv:2407.19583]
- B. Bašić, P. Ellis, D.C. Ernst, D. Popović, N. Sieben. Categories of impartial rulegraphs and gamegraphs. [arXiv:2312.00650]
- B.J. Benesh, D.C. Ernst, M. Meyer, S.K. Salmon, and N. Sieben. Impartial geodetic building games on graphs. [arXiv:2307.07095]

Below is a list of textbooks that I have written to be used with an inquiry-based learning (IBL) approach.

- D.C. Ernst and J. Hagood.
*Introduction to Discrete Mathematics*. Free undergraduate discrete mathematics textbook, 2024. [PDF] - D.C. Ernst.
*Introduction to Real Analysis*. Free undergraduate real analysis textbook, 2023. Modification of previous work: D.C. Ernst < K. Crisman < W.T. Mahavier < W.S. Mahavier. [PDF] - D.C. Ernst.
*An Inquiry-Based Approach to Abstract Algebra*. Open-source textbook for undergraduate abstract algebra, 2022. [Source] [PDF] - D.C. Ernst.
*An Introduction to Proof via Inquiry-Based Learning*. MAA Press, 2022. [Project Webpage] [Paperback]

- D.C. Ernst, J. Slye. Using the $\Spin_{3\times 3}$ virtual manipulative to introduce group theory.
*PRIMUS*34(6), 2024. [ePrint] - F. Awik, J. Breland, Q. Cadman, and D.C. Ernst. Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes.
*European Journal of Combinatorics*118, 2024. [arXiv:2104.12318] [ePrint] - B.J. Benesh, D.C. Ernst, and N. Sieben. The spectrum of nim-values for achievement games for generating finite groups.
*INTEGERS*23, 2023. [arXiv:2004.08980] [ePrint] - B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement games for generating nilpotent groups.
*J. Group Theory*22(3), 515–527, 2019. [arXiv:1805.01409] [ePrint] - D.C. Ernst. Diagram calculus for a type affine
*C*Temperley-Lieb algebra, II.*J. Pure Appl. Alg.*222(12), 3795-3830, 2018. [arXiv:1101.4215] [ePrint] - D.C. Ernst and N. Sieben. Impartial achievement and avoidance games for generating finite groups.
*Int. J. Game Theory*47(2), 509-542, 2018. [arXiv:1407.0784] [ePrint] - D.C. Ernst, T.J. Hitchman, and A. Hodge. Bringing Inquiry to the First Two Years of College Mathematics.
*PRIMUS*27(7), 641-645, 2017. [ePrint] - D.C. Ernst, A. Hodge, and S. Yoshinobu. Doceamus: What Is Inquiry-Based Learning?
*Notices of the AMS*64(6), 2017. [ePrint] - B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial achievement games for generating generalized dihedral groups.
*Australas. J. Combin.*68(3), 371-384, 2017. [arXiv:1608.00259] [ePrint] - D.C. Ernst, M. Hastings, and S. Salmon. Factorization of Temperley-Lieb diagrams.
*Involve*10(1), 89-108, 2017. [arXiv:1509.01241] [ePrint] - B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial avoidance and achievement games for generating symmetric and alternating groups.
*Int. Electron. J. Algebra*20, 70-85, 2016. [arXiv:1508.03419] [ePrint] - N. Diefenderfer, D.C. Ernst, M. Hastings, L.N. Heath, H. Prawzinsky, B. Preston, J. Rushall, E. White, A. Whittemore. Prime Vertex Labelings of Several Families of Graphs.
*Involve*9(4), 667-688, 2016. [arXiv:1503.08386] [ePrint] - B.J. Benesh, D.C. Ernst, and N. Sieben. Impartial avoidance games for generating finite groups.
*North-W. Eur. J. of Math.*2, 83-101, 2016. [arXiv:1506.07105] [ePrint] - H. Denoncourt, D.C. Ernst, and D. Story. On the number of commutation classes of the longest element of the symmetric group.
*Open Problems in Mathematics*4, 2016. [arXiv:1602.08328] [ePrint] - B. Beaudrie, D.C. Ernst, E. Kennedy, and R. St. Laurent. Inverted Pedagogy in Second Semester Calculus.
*PRIMUS*25(9-10), 992-906, 2015. [ePrint] - B. Love, A. Hodge, C. Corritore, and D.C. Ernst. Inquiry-Based Learning and the Flipped Classroom Model.
*PRIMUS*25(8), 745-762, 2015. [ePrtint] - D.C. Ernst, M. Leingang, and R. Taylor. Facebook for Professional Educators: To Friend or Not to Friend?
*MAA FOCUS*June/July 2015. [ePrint] - D.C. Ernst, A. Hodge, and A. Schultz. Enhancing Proof Writing via Cross-Institutional Peer Review.
*PRIMUS*25(2), 121-130, 2015. [ePrint] - D.C. Ernst. Diagram calculus for a type affine
*C*Temperley-Lieb algebra, I.*J. Pure Appl. Alg.*216(11), 2012. [arXiv:0910.0925] [ePrint] - T. Boothby, J. Burkert, M. Eichwald, D.C. Ernst, R.M. Green, and M. Macauley. On the cyclically fully commutative elements of Coxeter groups.
*J. Algebraic Combin.*36(1), 2012. [arXiv:1202.6657] [ePrint] - D.C. Ernst. Non-cancellable elements in type affine
*C*Coxeter groups.*Int. Electron. J. Algebra*8, 2010. [arXiv:0910.0923] [ePrint]

- D.C. Ernst and A. Hodge. Within \(\epsilon\) of Independence: An Attempt to Produce Independent Proof-Writers via IBL. In
*Beyond Lecture: Resources and Pedagogical Techniques for Enhancing the Teaching of Proof-Writing Across the Curriculum*, R. Schwell, A. Steurer, & J.F. Vasquez (Eds.), MAA Notes, 2016. - D.C. Ernst, A. Hodge, M. Jones, and S. Yoshinobu. The many faces of IBL. In
*STEM Education: An Overview of Contemporary Research, Trends, and Perspectives*, E. Ostler (Ed.), 2015. Elkhorn, NE.

- B. Beaudrie, D.C. Ernst, and B. Boschmans. Redesigning an Algebra for Precalculus Course. In
*Proceedings of World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education*, T. Bastiaens & G. Marks (Eds.), 2013. Chesapeake, VA: AACE. [EdITLib] - B. Beaudrie, B. Boschmans, and D.C. Ernst. First Semester Experiences in Implementing a Mathematics Emporium Model. In
*Proceedings of Society for Information Technology & Teacher Education International Conference*, R. McBride & M. Searson (Eds.), 2013. Chesapeake, VA: AACE. [EdITLib]

- D.C. Ernst.
*Instructor Guide: An Introduction to Proof via Inquiry-Based Learning*. [PDF] - D. Daly et al. AIBL Handbook for Online Professional Development: Lessons Learned from PRODUCT Workshops. Ethnography & Evaluation Research, & the Academy of Inquiry Based Learning. Boulder, CO, and San Luis Obispo, CA: University of Colorado Boulder, Ethnography & Evaluation Research; and Academy of Inquiry Based Learning. [ePrint]

From Spring 2013 through Spring 2016, I was co-editor and author for Math Ed Matters and from Fall 2016 through Fall 2017, was a co-editor and author for Teaching Tidbits. Both are online columns sponsored by the Mathematical Association of America. Below are several posts that I wrote for *Math Ed Matters*, *Teaching Tidbits*, as well as a few other online venues. I also write sporadically about mathematics and teaching on my personal blog.

- A conversation with Dana Ernst.
*IBL Teaching Podcast*. September 2023. [YouTube] - D.C. Ernst. The Role of Failure and Struggle in the Mathematics Classroom.
*Teaching Tidbits*. November 2017. [Blog Post] - D.C. Ernst. Want to Give Your Teaching Style a Makeover This Summer? Here’s How.
*Teaching Tidbits*. April 2017. [Blog Post] - D.C. Ernst. Who generates the examples?
*Teaching Tidbits*. November 2016. [Blog Post] - D.C. Ernst. Setting the Stage.
*Math Ed Matters*. January 2015. [Blog Post] - D.C. Ernst. The Twin Pillars of IBL.
*Math Ed Matters*. January 2015. [Blog Post] - D.C. Ernst. Fear is the mind-killer.
*Math Ed Matters*. June 2014. [Blog Post] - D.C. Ernst. Encouraging Students to Tinker.
*Math Ed Matters*. August 2014. [Blog Post] - D.C. Ernst, A. Hodge, and T.J. Hitchman. Engaging in Inquiry-Based Learning.
*Math Ed Matters*. February 2014. [Blog Post] - D.C. Ernst and A. Hodge. Math Ed Mania at the JMM.
*Math Ed Matters*. January 2014. [Blog Post] - D.C. Ernst and A. Hodge. The JMM: What’s Mathematics Education Got to Do with It?
*Math Ed Matters*. December 2013. [Blog Post] - D.C. Ernst. Give the Students the Colored Pen.
*Math Ed Matters*. August 2013. [Blog Post] - D.C. Ernst. Personality Matters?
*Math Ed Matters*. July 2013. [Blog Post] - D.C. Ernst. Grade School Utopia?
*Math Ed Matters*. July 2013. [Blog Post] - D.C. Ernst and A. Hodge. Try, Fail, Understand, Win.
*Math Ed Matters*. June 2013. [Blog Post] - D.C. Ernst. What the Heck Is IBL?
*Math Ed Matters*. May 2013. [Blog Post] - Teaching Calculus 1 with a Focus on Student Presentations.
*Discovering the Art of Mathematics*. October 2015. [Blog Post] - 4+1 interview with Dana Ernst.
*Casting Out Nines*by R. Talbert. The Chronicle Blog Network. August 2013. [Blog Post] - IBL Instructor Perspectives: Professor Dana Ernst.
*The IBL Blog*by S. Yoshinobu. February 2012. [Blog Post]

- D.C. Ernst. A diagrammatic representation of an affine
*C*Temperley-Lieb algebra, PhD Thesis, University of Colorado, 2008. [arXiv:0905.4457] - D.C. Ernst. Cell Complexes for Arrangements with Group Actions, MS Thesis, Northern Arizona University, 2000. [arXiv:0905.4434]

Below are all of the MS theses that I have advised at NAU.

- Hannah Golab. Pattern avoidance in Cayley permutations. Northern Arizona University MS Thesis. May 2024. [PDF]
- Ruth Perry. Braid graphs in Coxeter systems of type $\Lambda$ are median. Northern Arizona University MS Thesis. May 2024. [PDF]
- Jillian Barnes. Structural properties of braid graphs in simply-laced triangle-free Coxeter systems. Northern Arizona University MS Thesis. May 2022. [PDF]
- Fadi Awik. The reversal poset of signed permutations. Northern Arizona University MS Thesis. May 2021. [PDF]
- Quentin Cadman. Structure of braid graphs in simply-laced Coxeter systems. Northern Arizona University MS Thesis. May 2021. [PDF]
- Zachary Parker. On the maximum cardinality braid classes. Northern Arizona University MS Thesis. May 2017. [PDF]
- Kevin Salmon. Exploration of the type $\widetilde{C}$ Temperley-Lieb algebra. Northern Arizona University MS Thesis. May 2016. [PDF]
- Taryn Laird. A study of $T$-avoiding elements of Coxeter groups. Northern Arizona University MS Thesis. May 2016. [PDF]
- Kirsten N. Davis. A cellular quotient of the Temperley-Lieb algebra of type $D$. Northern Arizona University MS Thesis. May 2014. [PDF]
- Brooke Fox. Conjugacy of cyclically fully commutative elements in Coxeter groups of type $A$. Northern Arizona University MS Thesis. May 2014. [PDF]

- Dana Ernst: Math Professor, Race Across Arizona FKT Holder, Bikepacking Race Director.
*The Endurance Studio*. June 2024. [YouTube] - Dana Ernst: Verde Valley Randonneé Q&A.
*The Endurance Studio*. April 2024. [YouTube] - Bikepacking The Westfjords Way In Iceland: A Journey Through Europe’s Last Great Wilderness.
*The Seasoned Bikepaker*. July 2023. [YouTube] - M. Annabel, D.C. Ernst, C. Howard, W. Spalding. A Date with the Pines: Takes From the 2023 Pinyons and Pines.
*Bikepacking.com*. May 2023. [Article] - Strangely Alive? Arizona’s extreme endurance races.
*More Than a Game*. [Podcast] - Jesse Reeves and Dana Ernst on Winning the Stagecoach 400 Ultra Bikepacking Race.
*Stoke Podcast*. [Podcast] [YouTube] - Rekt: An interview with Dana Ernst, the Stagecoach 400 Ultra Endurance Bikepacking Event Co-Winner.
*The Seasoned Bikepaker*. [YouTube] - Dana Ernst and the Verde Valley Randonnee.
*Stoke Podcast*. [Podcast] [YouTube] - D.C. Ernst. 2019 Pinyons and Pines: Event Recap.
*Bikepacking.com*. August 2019. [Article] - D.C. Ernst. Dana’s AZT, Part 1.
*Bedrock Bags Blog*. June 2018. [Blog Post] - D.C. Ernst. Dana’s AZT, Part 2.
*Bedrock Bags Blog*. July 2018. [Blog Post]

Mathematics & Teaching

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Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.