## Profile

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### Dana Ernst

Shared publicly -Title: On the number of commutation classes of the longest element in the symmetric group

Original proposers of the open problem: Donald E. Knuth

The year when the open problem was proposed: 1992

Sponsor of the submission: Richard M. Green - University of Colorado Boulder

AMS Subject classification: 05

Status of the problem: Open

Abstract: Using the standard Coxeter presentation for the symmetric group S_n, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in S_n?

Thanks to +Richard Green for being our sponsor. Potentially of interest to +Drew Armstrong and +Christopher Hanusa.

### Dana Ernst

Shared publicly -https://plus.google.com/101584889282878921052/posts/gNmuwUfFDcU

In the second half of FAMUS, I discussed my path from hating mathematics as a child to falling in love with mathematics to eventually earning my PhD and becoming a professor of mathematics. I also shared a bit about what I love about mathematics, as well as the joys and struggles of teaching. The students seemed to really enjoy this.

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### Dana Ernst

Shared publicly -### Dana Ernst

Shared publicly -### Dana Ernst

Shared publicly -Title: Student presentations in calculus: A vehicle for getting students to talk about mathematics

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### Communities

18 communities### Dana Ernst

Shared publicly -I wasn't there, but I had some students present stuff.

Joni Hazelman and Parker Montfort presented the following poster.

Title: Explorations of Conway’s Sylver Coinage Game

Abstract: Sylver Coinage is a game in which two players, A and B, alternately name positive integers that are not the sum of nonnegative multiples of previously named integers. The person who names 1 is the loser! This seemingly innocent looking game is the subject of one of John Conway's open problems with monetary rewards. One such open problem is: If player A names 16 to start, and both players play optimally thereafter, then who wins? In this talk, we will discuss a simplified version of the game in which a fixed positive integer n (greater than 2) is agreed upon in advance. Then A and B alternately name positive integers from the set {1,2,...,n} that are not linear combinations with positive coefficients of previously named numbers. As in the original game, the person who is forced to name 1 is the loser. We will investigate who wins under optimal play for given values of n and determine the Nim-values for the simplified game under certain conditions. Joint work with Robert Voinescu and Ryan Wood.

Link: https://speakerdeck.com/dcernst/explorations-of-conways-sylver-coinage-game

+Hannah Paige Prawzinsky gave the following presentation.

Title: New coprime vertex labelings

Abstract: A coprime vertex labeling is an injective assignment of the labels {1, 2, . . . , n} to the vertices of an n-vertex simple connected graph such that adjacent vertices receive relatively prime labels. I will present new labelings for several infinite families of graphs. No prior knowledge of graph theory will be assumed. Joint work with Nathan Diefenderfer, Michael Hastings, Levi Heath, Briahna Preston, Emily White, and Alyssa Whittemore. This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

Link: https://speakerdeck.com/dcernst/new-coprime-vertex-labelings

### Dana Ernst

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### Dana Ernst

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### Dana Ernst

Shared publicly -**IBL Calculus and "Learn by Doing" Assignments**

This past fall I had the pleasure of teaching Calculus 1 to freshmen. It was a blast! I enjoyed every minute of it, and it's truly a privilege to be able to be one of the first professors that students see in college. Not only is it a great experience fo...

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### Dana Ernst

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### Dana Ernst

Shared publicly -**Quick Post: Re-Entering a New Term, Ken Robinson**

Happy New Year! Things have been relatively quiet on the IBL blog, but that's because we have some big things in the pipeline, such as a new NSF grant to expand IBL Workshop offerings. This is year 16 for me as a teacher (post PhD program). Year 20 includ...

- Northern Arizona UniversityAssistant Professor, 2012 - present
- University of Colorado at BoulderGraduate Student/Teaching Assistant, 2003 - 2008
- Front Range Community CollegeMath Faculty, 2001 - 2003
- Northern Arizona UniversityGraduate Student/Instructor, 1997 - 2001
- Plymouth State UniversityAssistant Professor, 2008 - 2012

- Dana's Blog (current)
- Math Ed Matters (current)
- Teaching Tidbits (current)

My name is Dana Ernst and I am an assistant professor in the Department of Mathematics and Statistics at Northern Arizona University in Flagstaff, AZ.

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. 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. See my scholarship page for more information.

Furthermore, I am passionate about mathematics education. In particular, I am interested in inquiry-based learning (IBL) and the Moore method for teaching mathematics. This educational paradigm has transformed my teaching. 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.

I am also interested in utilizing technology to enhance the teaching and learning of mathematics. Specifically, I choose free and open-source software and technologies when appropriate. For example, I have been incorporating Sage and GeoGebra into my teaching. Sage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface. For examples of a few of the cool things you can do with Sage, check this page. According to their webpage, GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. There are tons of awesome GeoGebra examples located here.

In addition to using free and open-source software, I am inspired by the recent open-source textbookmovement and I strongly believe that educators should choose free, open-source, or low cost textbooks when a viable alternative exists. For a list of open-source textbooks, go here and here.

Angie Hodge and I are coauthors for Math Ed Matters, which is a (roughly) monthly column sponsored by the Mathematical Association of America. The column explores topics and current events related to undergraduate mathematics education. Posts aim to inspire, provoke deep thought, and provide ideas for the mathematics—and mathematics education—classroom. Our interest in and engagement with IBL color the column's content.

I also maintain a personal blog, which is part of the Booles' Rings network of academic home pages/blogs. On my blog, I typically write about topics related to mathematics, education, and technology. In addition, I occasionally post about my cycling, trailing running, and rock climbing adventures on my Elevation Gain blog.

Lastly, I am a husband and a father of two incredible sons. Oh, I enjoy drinking copious amounts of coffee, too.

- George Mason UniversityBS, Mathematics, 1993 - 1997
- Northern Arizona UniversityMS, Mathematics, 1997 - 2000
- University of Colorado at BoulderPhD, Mathematics, 2003 - 2008