I am an assistant professor here in the Department of Mathematics at Bucknell University in central Pennsylvania. In 2003, I received my Ph.D. from MIT, where Richard Stanley was my advisor. I was then a postdoc for two years with the Laboratoire de Combinatoire et d'Informatique Mathématique (LaCIM), located at the Université du Québec à Montréal, where my mentors were François Bergeron and Christophe Reutenauer. Immediately before starting at Bucknell, I was postdoc for one year at the Center for Mathematical Analysis, Geometry and Dynamical Systems, in the Department of Mathematics at the Instituto Superior Técnico in Lisbon.

My mathematical interests are in combinatorics, often of an algebraic or order-theoretic nature. Some of my recent work has had to do with P-partitions, edge-labellings of partially ordered sets, Schur-positivity, and relationships among skew Schur functions.

Papers and Publications

• New! With Bruce Sagan: "Infinite log-concavity: developments and conjectures," preprint. (pdf, ps, ps.gz, or read the abstract.)

• With Stephanie van Willigenburg: "Positivity Results on Ribbon Schur Function Differences," preprint. (pdf, ps, ps.gz, or read the abstract.)

• "Necessary Conditions for Schur-Positivity," Journal of Algebraic Combinatorics, to appear. (pdf, ps, ps.gz, or read the abstract.)

• With Stephanie van Willigenburg: "Towards a Combinatorial Classification of Skew Schur Functions," Transactions of the American Mathematical Society, to appear. (pdf, ps, ps.gz, or read the abstract.)

• With Christophe Reutenauer: "P-Partitions and a Multi-Parameter Klyachko Idempotent," Electronic Journal of Combinatorics 11(2) (2005), #R21, 18pp. Special volume in honor of Richard Stanley on the occasion of his 60th birthday. (pdf, ps, or read the abstract.)

• With François Bergeron: "Some Positive Differences of Products of Schur Functions." (pdf, ps, ps.gz, or read the abstract. There is also an update.)

• With Charles R. Johnson, Stefan Leichenauer and Roberto Costas: "Principal Minor Sums of (A+tB)^m," Linear Algebra and its Applications, 411 (2005), 386-389. Special issue on determinants and the legacy of Sir Thomas Muir. (pdf (Recommended), ps, or read the abstract.) This article stemmed from Leichenauer's final report at the Research Experience for Undergraduates program during Summer 2004 at the College of William and Mary, Williamsburg, Virginia. Costas, Johnson and I served as advisors.

• "Cylindric Skew Schur Functions," Advances in Mathematics, 205 (1) (2006), 275-312. (pdf, ps, ps.gz, or read the abstract.)

• "Edge Labellings of Partially Ordered Sets," Ph.D. thesis, MIT, 2003. (pdf, ps, ps.gz or read the abstract. There is also an update.)

• With Hugh Thomas: "Poset Edge-Labellings and Left Modularity," European Journal of Combinatorics 27 (2006), 101-113. (pdf, ps or read the abstract.) A bonus section not included in the published version introduces the non-straddling partitions of [n], a subset of the non-nesting partitions of [n].

• "EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets," Journal of Combinatorial Theory (Series A) 101 (2003), 69-89. (pdf, ps, ps.gz or read the abstract.)

Talks

• New! "The Schur-Positivity Poset," AMS Special Session on Combinatorics of Partially Ordered Sets, AMS Western Section Meeting, Claremont, California, 3rd May 2008. (Slides in pdf, ps, ps.gz, or read the abstract.)

• "Tilings from the Floor Up," Student Colloquium Series, Bucknell University, 25th October 2007. (Slides in pdf (1.2 MB), ps (7.8 MB), ps.gz (1.2 MB), or read the abstract.) This was a talk aimed at undergraduates and was based on an expository paper of Federico Ardila and Richard Stanley.

• "The Schur-Positivity Poset," Algebra etc. Seminar, Bucknell University, 20th September 2007. This was the second of two talks, the first of which was an introduction to symmetric functions and, in particular, skew Schur functions and the Littlewood-Richardson rule. (Slides in pdf, ps, ps.gz, or read the abstract.) Some posets were displayed on the blackboard during the talk; pages displaying these posets have been incorporated into the slides files.

• "A Combinatorial Classification of Skew Schur Functions," Formal Power Series and Algebraic Combinatorics, Tianjin, China, 3rd July 2007. (Slides in pdf (recommended), ps, ps.gz, or read the abstract.)

• "A Combinatorial Classification of Skew Schur Functions," Workshop on Combinatorial Hopf Algebras and Macdonald Polynomials, Centre de Recherches Mathématiques, Montreal, 11th May 2007. (Slides in pdf, or read the abstract.)

• "A Combinatorial Classification of Skew Schur Functions," AMS Special Session on Algebraic Combinatorics, Southeastern Section Meeting, Fayetteville, Arkansas, 3rd November 2006. (Slides in pdf, ps, or read the abstract.)

• "When Are Two Skew Schur Functions Equal?" University of Michigan Combinatorics Seminar, 22nd September 2006. (Slides in pdf (Recommended), ps, or read the abstract.)

• "A Combinatorial Lie Element for the Symmetric Group Algebra," Algebra Seminar, Universidade de Coimbra, Portugal, 29th November 2005. (A chalk talk. Read the abstract.)

• "Symmetric Functions and Cylindric Schur Functions," Seminário do Centro de Análise, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, 11th October 2005. (Slides in pdf, ps, or read the abstract.)

• "Positivity Questions for Cylindric Skew Schur Functions," Formal Power Series and Algebraic Combinatorics, Taormina, Sicily, Italy, 21st June 2005. (Slides in pdf, ps, ps.gz or read the abstract.)

• "An Introduction to the Combinatorics of Symmetric Functions," Pomona College, 4th February 2005. (Slides in pdf, ps, or read the abstract.)

• "Symmetric Functions and Cylindric Schur Functions," Bucknell University, 31st January 2005. (Slides in pdf or ps.)

• "Cylindric Skew Schur Functions," University of Minnesota Combinatorics Seminar, 5th November 2004. (Slides in pdf (Recommended), ps, or read the abstract.)

• "Positivity Questions for Generalised Schur Functions," Le Séminaire de Combinatoire et d'Informatique Théorique du LaCIM, 1st October 2004. (Slides in pdf (Recommended), ps, or read the abstract.)

• "Cylindric Schur Functions," Retrospective in Combinatorics: Honoring Stanley's 60th Birthday, MIT, 24th June 2004. (Slides in pdf (Recommended), ps, or read the abstract.)

• "P-partitions and Quasi-Symmetric Functions," York University Applied Algebra Seminar, 3rd February 2004. (Slides in pdf (Recommended), ps, or read the abstract. There is also an update.)

• "Questions of Schur-Positivity," McGill Discrete Mathematics and Optimization Seminar, 10th November 2003. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Edge Labellings of Partially Ordered Sets and Their Implications," Le Séminaire de Combinatoire et d'Informatique Théorique du LaCIM, 12th September 2003. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Poset Edge-Labellings and Left Modularity," Formal Power Series and Algebraic Combinatorics, Vadstena, Sweden, 27th June 2003. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Edge Labellings of Partially Ordered Sets," Ph.D. thesis defense, 25th April 2003. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Equivalent characterizations of lattice supersolvability and their extensions," AMS Special Session on The Many Lives of Lattice Theory and the Theory of Ordered Sets, with Connections to Combinatorics, Joint Mathematical Meetings, Baltimore, 18th January 2003. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Permutation Edge-Labellings of Partially Ordered Sets," Canadian Mathematical Society Summer Meeting, Québec City, 17th June 2002. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Two New Characterizations of Lattice Supersolvability," MIT Combinatorics Seminar, 5th December 2001. (Slides in pdf, ps, ps.gz or read the abstract.)

• "Let's talk about posets, baby," Simple Person's Applied Math Seminar, 17th October 2001. (Slides in pdf, ps, ps.gz or read the abstract. Handout in pdf or ps.) A very similar talk with a less goofy title was given at the Dartmouth Graduate Student Seminar, 16th April 2002.

Research related material

• Research Statement (2004) (pdf, ps)

• Special Session in Algebraic Combinatorics, CMS Winter Meeting 2004.

Related Links

• The Math Genealogy Project's results for Richard Stanley
• Richard Ehrenborg's page with Gian-Carlo Rota's academic descendants
• Mark Skandera's Combinatorics photo album
• The other mathematical Peter McNamara, a Ph.D. student at MIT