My primary research interests are

• Algebraic and enumerative combinatorics, particularly connections with permutations, tableaux, and representation theory;
• Dynamical algebraic combinatorics, where one adds the ingredient of a group action to the above mix. This includes the cyclic sieving phenomenon and the homomesy phenomenon.
• Mathematics education, particularly outreach to K–12 teachers; support for K–16 learners; and online and distance learning tools (e.g., lightboard video lectures). Here is complete set of lightboard video lectures I made for our Applied Linear Algebra course.
  • A pdf version of my CV
  • Schensted Algorithm demos by my former masters student Dave Goggin
  • Webpage of my former doctoral student Michael Joseph

  Mathematics Publications and Preprints:

  • Birational rowmotion on a rectangle over a noncommutative ring (with Darij Grinberg) arXiv preprint
    
    
  • Foatic actions of the symmetric group and fixed-point homomesy (with Michael LaCroix), arXiv preprint.
    
    
  • A birational lifting of the Stanley–Thomas word on products of two chains (with Michael J. Joseph), submitted.
    
    
  • Birational and noncommutative lifts of antichain toggling and rowmotion (with Michael J. Joseph) to appear in Algebraic Combinatorics.
    
    
  • Paths to Understanding Birational Rowmotion on Products of Two Chains (with Gregg Musiker), Algebraic Combinatorics. 2 #2 (2019), pp. 275–304.
    Poster version presented at FPSAC 2018
    
    
  • Whirling injections, surjections, and other functions between finite sets (with Michael J. Joseph and James Propp), preprint.
    
    
  • Toggling independent sets of a path graph (with Michael J. Joseph) in Electronic Journal of Combinatorics, Volume 25, Issue 1 (2018), Paper #P1.18.
    
    
  • Dynamical Algebraic Combinatorics and the Homomesy Phenomenon in A.~Beveridge, J.~Griggs, L.~Hogben, G.~Musiker, P. Tetali, eds., Recent Trends in Combinatorics (IMA Volume in Mathematics and its Applications), Springer, 2016.
    
    
  • Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets (with Darij Grinberg) in Electronic Journal of Combinatorics, Volume 23, Issue 1 (2016), Paper #P1.33.
    
    
  • Iterative Properties of Birational Rowmotion II: Rectangles and Triangles (with Darij Grinberg) in Electronic Journal of Combinatorics, Volume 22, Issue 3 (2015), > Paper #P3.40.
    
    
  • Homomesy in Products of Two Chains (with James Propp), in Electronic Journal of Combinatorics, Volume 22, Issue 3 (2015), Paper #P3.4.
    
    
  • Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions (with S. Linton, J. Propp, J. West), Journal of Integer Sequences 15 (2012), Article 12.9.1.
    
    
  • Equivalence Relations of Permutations Generated by Constrained Transpositions (with S. Linton, J. Propp, J. West), Disc. Math. & Theoretical Comp. Sci. Proceedings, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010.
    
    
  • A two-dimensional pictorial presentation of Berele's insertion algorithm for symplectic tableaux (with I. Terada). Electronic Journal of Combinatorics, Volume 12(1), 2005, R4.
    
    
  • The Lattice of Threshold Graphs (with Russ Merris), J. Ineq. Pure & Appl. Math (JIPAM) vol. 6, #1 (2005)
    
    
  • A Schensted insertion for tensor powers of the Weil representation, accepted as a poster at FPSAC 2002, Melbourne.
    
    
  • Complementary Algorithms for Tableaux (with Frank Sottile, Jeff Stroomer, and Julian West). Journal of Combinatorial TheoryA 96, 127--161 (2001). Available in postscript or tarred and gzipped .
    
    
  • Jeux de tableaux (With F. Sottile, J. Stroomer, J. West). Formal Power Series and Algebraic Combinatorics: Twelfth International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings, pp. 332--343.
    
    
  • Down-up Algebras (with Georgia Benkart) Journal of Algebra 209 , 3-5--344 (1998) A brief addendum is available.
    
    
  • The connection between the Robinson-Schensted correspondence for skew oscillating tableaux and graded graphs. Discrete Math. 139 (1995) 481--485.
    
    
  • Applications and Extensions of Fomin's Generalization of the Robinson-Schensted Correspondence to Differential Posets, Ph.D. Thesis, Massachusetts Institute of Technology, 1991. Here's a PDF scan of this.
    
    

  Slides from talks (reverse chronologically):

  NB: The animations by Mike LaCroix in several of the presentations below run (only?) in versions of adobe acrobat (reader), not in preview, but summary slides viewable in other readers are included for each animation.
  
  
  • Rowmotion orbits in finite poset, given at the Conference on Enumerative and Algebraic Combinatorics honoring Bruce Sagan at Department of Mathematics, University of Florida, Gainesville, FL, 27 February 2024.
    
    
  • Jeu de Taquin: The Fifteen Puzzle in Research Mathematics, given at the 15th Gathering for Gardner Atlanta, GA 22 February 2024.
    
    
  • Dynamical algebraic combinatorics: Actions, orbits, and averages, given at the Math Department Colloquium at Tufts University, Medford, MA, 2 February 2024.
    
    
  • Lifting Rowmotion to higher realms and noncommutative periodicity, given at the Dartmouth College Combinatorics Seminar, 23 May 2023.
    
    
  • Dynamical Algebraic Combinatorics: Actions, Orbits, and Averages, given at the Max Planck Institut für Mathematik in den Naturwissenschaften Leipzig, Germany, 14 February 2023.
    
    
  • Rowmotion in products of two chains, JMM abstract link given at the AMS Special Session on The Many Lives of Lattice Theory with an Emphasis on Distributive & Semi-distributive Lattices and Combinatorics at the 2022 Joint Mathematics Meetings, 7 April 2022.
    
    
  • Some recent orbits of homomesy, given at the Dartmouth College Combinatorics Seminar, 1 March (Mardi Gras) 2022.
    
    
  • Dynamical algebraic combinatorics in 10 Minutes, A short introduction given at UConn Algebra Seminar, 9 September 2020.
    
    
  • Dynamical algebraic combinatorics and homomesy: An action-packed introduction , given at AlCoVE: an Algebraic Combinatorics Virtual Expedition, 15–16 June 2020. VIDEO AVAILABLE
    
    
  • Dynamical Algebraic Combinatorics: Actions, Orbits, and Averages, given at the Program in Algebraic and Enumerative Combinatorics at the Institut Mittag-Leffler, Djursholm, SWEDEN, 25 February 2020.
    
    
  • Dynamical Algebraic Combinatorics: Actions, Orbits, Averages, Sieving, given in the Graduate School of Mathematics, at Nagoya University, Japan, 7 May 2019.
    
    
  • Paths to Understanding Rowmotion on a Product of Two Chains, given in the Department of Mathematics, at Okayama University, Japan, 28 March 2019.
    
    
  • Dynamical Algebraic Combinatorics and the Homomesy Phenomenon: Toggling, whirling, and Bulgarian solitaire given in the Department of Mathematics, at Okayama University, Japan, 26 March 2019.
    
    
  • Paths to Understanding Rowmotion on a Product of Two Chains, given in the SageDays@ICERM: Combinatorics and Representation Theory, at The Institute for Computational and Experimental Research in Mathematics at Brown University, 25 July 2018.
    
    
  • Dynamical Algebraic Combinatorics: Actions, Orbits, and Averages, given in the Discrete Math Seminar, at UMassAmherst Dept. of Math & Statistics, Amherst, MA 26 April 2018.
    
    
  • Paths to Understanding Rowmotion on a Product of Two Chains, given in the Combinatorics Seminar, at Brandeis University (Waltham, MA), 17 April 2018.
    
    
  • Dynamical Algebraic Combinatorics: Actions, Orbits, and Averages, given in the NDSU Math Colloquium, at North Dakota State University, Fargo, 27 March 2018.
    
    
  • Dynamical Algebraic Combinatorics and the Homomesy Phenomenon on Independent Sets of a Path Graph and on Permutations , given at Tsuda University, Tokyo, Japan, 2 March 2018.
    
    
  • Paths to Understanding Birational Rowmotion on a Product of Two Chains, given in the Graduate School of Mathematics, at Nagoya University (Japan), 5 March 2018.
    
    
  • Paths to Understanding Birational Rowmotion on a Product of Two Chains, given in the conference Algebraic and Enumerative Combinatorics in Okayama, at < a href="https://www.gnst.okayama-u.ac.jp/en/">Graduate School of Natural Science and Technology at Okayama University (Japan), 22 February 2018.
    
    
  • Dynamical Algebraic Combinatorics and the Homomesy Phenomenon, given in the WPI Discrete Math Seminar, at Worcester Polytechnique Institute, 8 February 2018.
    
    
  • A (lattice) path formula for birational rowmotion on a product of two chains, given in the Workshop on Algorithmic and Enumerative Combinatorics as part of the Programme on Algorithmic and Enumerative Combinatorics held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), 15 November 2017.
    
    
  • Homomesies Lurking in the Twelvefold Way, given in the Special Session on Enumerative Combinatorics held at the AMS Central Sectional Meeting, 28 October 2016.
    
    
  • Homomesies Lurking in the Twelvefold Way, given in the Special Session on Algebraic and Enumerative Combinatorics held at the AMS Eastern Sectional Meeting at Bowdoin College, 25 September 2016.
    
    
  • Birational Rowmotion: order, homomesy, and cluster connections, given in the Combinatorics Seminar, at University of Minnesota, 22 May 2015.
    
    
  • Birational Rowmotion, given at (the NSF co-sponsored) Dynamical Algebraic Combinatorics Workshop at American Institute of Mathematics, San Jose, CA, 24 March 2015.
    
    
  • See-Saw Swap Solitaire and Other Games on Permutations, given at Canada/USA Mathcamp, Lewis & Clark, Portland, OR, 29 July 2014.
    
    
  • Rowmotion: Classical \& Birational, given at Stanley@70 Conference, MIT, Cambridge, MA, 26 June 2014.
    
    
  • The Eleven Clocks Problem, given at Gathering for Gardner XI, Atlanta, GA, 22 March 2014.
    
    
  • Homomesy of Order Ideals in Products of Two Chains given in the Combinatorics Seminar, at University of Minnesota, 6 Sept 2013.
    
    
  • (with Darij Grinberg) Combinatorial Actions, Orbit Averages, and Sage Implementation given at Sage Days 49, 18 June 2013.
    
    
  • Combinatorial Actions and Homomesic Orbit Averages given in the Mathematics Colloquium, at Dartmouth College, 23 May 2013.
    
    
  • Homomesy of Order Ideals in Products of Two Chains given in the Combinatorics Seminar, at Dartmouth College, 23 May 2013.
    
    
  • Combinatorial Ergodicity given in the Algebra seminar, at the University of Connecticut, 27 March 2013.
    
    
  • Combinatorial Ergodicity given in the Workshop on Algebraic Combinatorics related to Young diagrams and Statistical Physics, at the International Institute for Advanced Study in Kizugawa-shi, Kyoto, Japan, 9 August 2012.
    
    
  • Combinatorial Ergodicity given at the University of Tokyo, Department of Mathematical Sciences, Tokyo, Japan, 23 July 2012.
    
    
  • Combinatorial Ergodicity in Products of Chains given at SIAM Discrete Math 2012 Conference at Dalhausie University, Halifax, NS, Canada, 18 June 2012.
    
    
  • See-Saw Swap Solitaire, given at Gathering for Gardner X, Atlanta, GA, 29 March 2012.
    
    
  • A Bijection that Counts: The Robinson-Schensted-Knuth Correspondence at Joint Math Meetings, 5 January 2012.
    
    
  • Equivalences relations of permutations generated by constrained transpositions in MIT Combinatorics Seminar, 18 November 2011.
    
    
  • More history to appear eventually...!

  Math Education Publications and Preprints:

  • More recent texts to appear soon...
    
    
  • (with Maletsky, E. and seven other authors) Harcourt Math (Grades K--6) Harcourt, Inc., Orlando, 2004.
    
    
  • "Models for fractions" in Harcourt Math Professional Handbook (Orlando, 2002), pp. PH24--25.
    
    
  • The JOMA mathlets project. Journal of Online Mathematics and its Applications 1 (Jan, 2001).