Eric S. Egge's Home Page

I am originally from Park Rapids, Minnesota, a small resort town in the lake country of central Minnesota. I did my undergraduate work in Mathematics at Carleton College in Northfield, Minnesota. I did my graduate work in Mathematics at the University of Wisconsin-Madison, under the direction of Paul Terwilliger. I earned my Ph. D. in Mathematics in May of 2000 and then spent five years as an Assistant Professor of Mathematics at Gettysburg College. I returned to Carleton in the fall of 2005, as an Assistant Professor of Mathematics, and I was promoted to Associate Professor of Mathematics in the fall of 2009.

My mathematical interests are in algebra and combinatorics, especially enumerative and algebraic combinatorics. My research includes work on pattern-avoiding permutations, a generalization of the Terwilliger algebra of a distance-regular graph or association scheme, the combinatorics of symmetric functions, and the combinatorics of the Legendre-Stirling and Jacobi-Stirling numbers. I am also interested in undergraduate problem-solving, undergraduate research, and games and puzzles with mathematical connections. In my spare time I enjoy reading, writing, running, baking pizza, cheesecake, cookies, and brownies, and playing games with my family.

Each electronic version below is my copy of the final version of the listed paper. To obtain a copy of the published version of any paper for which an electronic version is listed, please contact me directly.

The Jacobi-Stirling Numbers, with George E. Andrews, Wolfgang Gawronski, and Lance L. Littlejohn, Journal of Combinatorial Theory, Series A, accepted, preprint on arXiv.

From Quasisymmetric Expansions to Schur Expansions via a Modified Inverse Kostka Matrix, with Nicholas A. Loehr and Gregory S. Warrington, European Journal of Combinatorics, vol. 31, pp. 2014-2027, 2010, electronic version.

Legendre-Stirling Permutations, European Journal of Combinatorics, vol. 31, pp. 1735-1750, 2010, electronic version.

Enumerating rc-Invariant Permutations with No Long Decreasing Subsequences, Annals of Combinatorics, vol. 14, pp. 85-101, 2010, electronic version.

The Pfaffian Transform, with Tracale Austin, Hans Bantilan, Isao Jonas, and Paul Kory, Journal of Integer Sequences, vol. 12, Article 09.1.5, 2009, published version.

Restricted Symmetric Permutations, Annals of Combinatorics, vol. 11, pp. 405-434, 2007, electronic version.

Restricted Colored Permutations and Chebyshev Polynomials, Discrete Mathematics, vol. 307, pp. 1792-1800, 2007, electronic version.

Restricted Signed Permutations Counted by the Schroder Numbers, Discrete Mathematics, vol. 306, pp. 552-563, 2006, electronic version.

Bivariate Generating Functions for Involutions Restricted by 3412, with Toufik Mansour, Advances in Applied Mathematics, vol. 36, pp. 118-137, 2006, electronic version.

Restricted Permutations, Fibonacci Numbers, and k-Generalized Fibonacci Numbers, with Toufik Mansour, Integers: The Electronic Journal of Combinatorial Number Theory, vol. 5, article A1, 2005, online version.

231-Avoiding Involutions and Fibonacci Numbers, with Toufik Mansour, Australasian Journal of Combinatorics, vol. 30, pp. 75--84, 2004, electronic version.

Restricted 3412-Avoiding Involutions, Continued Fractions, and Chebyshev Polynomials, Advances in Applied Mathematics, vol. 33, pp. 451-475, 2004, electronic version.

132-Avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, with Toufik Mansour, Discrete Applied Mathematics, vol. 143, pp. 72-83, 2004, electronic version.

A Schroder Generalization of Haglund's Statistic on Catalan Paths, with James Haglund, Kendra Killpatrick, and Darla Kremer, Electronic Journal of Combinatorics, vol. 10, article #R16, 2003, online version.

Permutations Which Avoid 1243 and 2143, Continued Fractions, and Chebyshev Polynomials, with Toufik Mansour, Electronic Journal of Combinatorics, vol. 9(2), article #R7, 2003, online version.

A Weight-Preserving Bijection Between Schroder Paths and Schroder Permutations, with Jason Bandlow and Kendra Killpatrick, Annals of Combinatorics, vol. 6, pp. 235-248, 2002, electronic version.

The Generalized Terwilliger Algebra and its Finite Dimensional Modules when d=2, Journal of Algebra, vol. 250, pp. 178-216, 2002, electronic version.

A Generalization of the Terwilliger Algebra, Journal of Algebra, vol. 233, pp. 213-252, 2000, electronic version.

Here are links to the slides for some of my recent talks.

New Pattern-Avoiding Permutations Counted by the Schroder Numbers, AMS Special Session on Permutation Patterns, Algorithms, and Enumerative Combinatorics, Rochester, NY, September 23, 2012, slides.

Pattern-Avoiding Permutations and Lattice Paths: Old Connections and New Links, MAA Mathfest Invited Paper Session Walk the Walk, Talk the Talk, Madison, WI, August 3, 2012, slides.

Linear Recurrences and the Pfaffian Transform, MAA Mathfest Invited Paper Session on Combinatorics and Matrices, Madison, WI, August 2, 2012, slides.

The Jacobi-Stirling Numbers, AMS Special Session on Symmetric Functions, Quasisymmetric Functions, and Associated Combinatorics, George Washington University, Washington, DC, March 18, 2012, slides.

Symmetric Permutations with No Long Decreasing Subsequences, Joint Mathematics Meetings, AMS Session on Discrete Mathematics, San Francisco, CA, January 15, 2010, slides.

Here are several of the student projects I have supervised over the past seven years. Where possible, I have included links to papers in which the students wrote up their results. In other cases I have given a brief description of the project.

Restricted Symmetric Signed Permutations, Andy Hardt '13 and Justin Troyka '13, Summer 2011, results are available in a preprint submitted for publication.

A q=-1 Phenomenon for Pattern-Avoiding Permutations, Xin Chen '13, December 2010, results published in Rose-Hulman Undergraduate Mathematics Journal, volume 12, issue 2, 2011, online version.

A Higman-Sims Puzzle, Erica Chesley '10, Zack Starer-Stor '10, and Emma Zhou '10, Winter and Spring 2010.

It is well known that permutation puzzles like Loyd's 15-puzzle and the Rubik's cube each have an associated group. In most cases this group is the set of permutations of the elements of the puzzle which are accessible via sequences of legal puzzle moves (no fair removing stickers!) but in the case of the 15-puzzle we only consider those configurations in which the blank is in a given, fixed, position. Generally speaking, people design these puzzles for fun, and only later is the group of the puzzle determined. In an article in the July 2008 issue of Scientific American, Igor Kriz and Paul Siegel turned this paradigm around: instead of designing puzzles and then determining the associated groups, they chose finite simple groups and designed associated puzzles. In particular, Kriz and Siegel designed puzzles for the Mathieu groups M₁₂ and M₂₄, as well as for the Conway group Co₀. Inspired by Kriz and Siegel's work, in the winter and spring of 2010 math majors Erica Chesley, Zack Starer-Stor, and Emma Zhou set out to do the same thing for other finite simple groups. They considered many puzzles and a variety of groups, and eventually designed a beautiful puzzle for the Higman-Sims group, a finite simple group of order 44,352,000 which was discovered by Higman and Sims in the late 1960s. This puzzle is available for the ipad in the Apple app store.

Legendre-Stirling Number Identities, Alex Fisher '10, Summer 2009.

In 2002 Everitt, Littlejohn, and Wellman introduced the Legendre-Stirling numbers in connection with a differential operator related to Legendre polynomials. As their name suggests, the Legendre-Stirling numbers of the first and second kinds generalize the Stirling numbers of the first and second kinds, which count permutations according to length and number of cycles and set partitions according the number of elements and number of blocks, respectively. In 2008 Andrews and Littlejohn gave a combinatorial interpretation of the Legendre-Stirling numbers of the second kind in terms of a certain type of set partition, and in early 2009 I gave a combinatorial interpretation of the Legendre-Stirling numbers of the first kind in terms of pairs of permutations. In the summer of 2009 Alex Fisher used these two combinatorial interpretations to give combinatorial proofs of some identities involving Legendre-Stirling numbers, which generalize identities involving Stirling numbers. Alex also took this work a step further, proving generalizations of these identities for a wide array of number triangles whose entries satisfy a recurrence similar to that of Pascal's triangle. Alex presented his results at the annual Pi Mu Epsilon student conference at St. John's University in April of 2010. You can see a poster describing Alex's results here.

Harmonic Functions on Young's Lattice, Long Chan '11 and Erin Jones '12, Summer 2009, results will appear this fall in the Pi Mu Epsilon Journal, and are available in an electronic version.

Alternating Sign Matrices, Nathan Williams '08, Spring 2008, results published in Rose-Hulman Undergraduate Mathematics Journal, volume 9, issue 2, 2008, online version.

Kepler Towers, Kepler Walls, and Narayana Statistics, Adrian Duane '07, Spring 2007 to Spring 2008, results available in a preprint.

Symmetric Pattern-Avoiding Permutations, David Lonoff '09 and Jonah Ostroff '08, Summer 2007, results published in Annals of Combinatorics, vol. 14, pp. 143-158, 2010, and are available in an electronic version.

The Pfaffian Transform, Tracale Austin '07, Hans Bantilan '07, Isao Jonas '07, and Paul Kory '07, Fall 2006 and Winter 2007, results are published in Journal of Integer Sequences, vol. 12, Article 09.1.5, 2009, online version.