Welcome to Tom Roby's Research Page!
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.
Mathematics Publications and Preprints:
 Foatic actions of the symmetric group and fixedpoint 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 twodimensional 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, 127161 (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. 332343.

Downup Algebras (with Georgia Benkart) Journal of Algebra 209
, 35344 (1998) A brief addendum is
available.
 The connection between the RobinsonSchensted
correspondence for skew oscillating tableaux and graded graphs.
Discrete Math. 139 (1995) 481485.
 Applications and Extensions of Fomin's
Generalization of the RobinsonSchensted 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.
 Dynamical algebraic combinatorics in 10 Minutes, A short
introduction given at UConn Algebra Seminar, 9 September 2020.
 Dynamical algebraic combinatorics and homomesy: An actionpacked 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 MittagLeffler,
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.okayamau.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 SchrÃ¶dinger 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 cosponsored)
Dynamical Algebraic Combinatorics Workshop at
American Institute of Mathematics, San Jose, CA, 24 March
2015.
 SeeSaw 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 Kizugawashi, 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.
 SeeSaw Swap
Solitaire, given
at Gathering for
Gardner X, Atlanta, GA, 29 March 2012.
 A Bijection that Counts: The
RobinsonSchenstedKnuth 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 K6) Harcourt, Inc., Orlando, 2004.
 "Models for fractions" in Harcourt Math Professional
Handbook (Orlando, 2002), pp. PH2425.
 The JOMA mathlets project. Journal
of Online Mathematics and its Applications 1 (Jan, 2001).
