Publications

Clicking here will take you to all the papers I have posted on the arXiv system (all but my earliest two papers are there). Below, I have given direct links to my papers and in each case the link is to the version I think is best (not always the version on the arxiv).

Preprints

• Enumeration of noncrossing pairings on bit strings (with T. Kemp, K. Mahlburg and C. Smyth)
  

  Abstract: A non-crossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings we generalize classical problems from the theory of Catalan structures. In particular, it is very difficult to find useful explicit formulas for the enumeration function which counts the number of pairings as a function of the underlying bitstring. We determine explicit formulas for enumeration function for bit strings, and also prove general upper bounds in terms of Fuss-Catalan numbers by relating non-crossing pairings to other generalized Catalan structures (that are in some sense more natural). This enumeration problem arises in the theory of random matrices and free probability.

Refereed Publications

• The number of lattice paths below a cyclically shifting boundary (with J. Irving)
  J. Combin. Theory Ser. A, 116(3):499--514, April 2009

  Abstract: We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical reflection argument. A refinement allows for the counting of paths with a specified number of corners. We also apply the result to examine paths dominated by periodic boundaries.

• Minimal factorizations of permutations into star transpositions (with J. Irving)
  Discrete Math, 309(6):1435--1442, April 2009.

  Abstract: We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form (1 i). Our result generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored.

• Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula (with P. Śniady)
  Adv. Math, 218(3):673--695, 2008

  Abstract: We study asymptotics of an irreducible representation of the symmetric group S_n corresponding to a balanced Young diagram λ (a Young diagram with at most O() rows and columns) in the limit as n tends to infinity. We find an asymptotic bound for characters χ^λ(π). Our main achievement is that - contrary to previous results in this direction - we do not assume that the length ∣π∣ of the permutation is small in comparison to n. The main tool we introduce is a generalization of the Frobenius character formula, which holds true not only for cycles but for arbitrary permutations.

• Stanley’s character polynomials and coloured factorizations in the symmetric group
  J. Combin. Theory Ser. A, 115(4):535--546, May 2008.

  Abstract: Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. He later gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.

• Positivity results for Stanley’s character polynomials
  J. Algebra, 308(1):26-47, Feb. 2007.

  Abstract: Stanley introduced expressions for the normalized characters of the symmetric group and stated some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley’s positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results by Goulden and Rattan. We shall see that the generating series C(t) introduced by Goulden and Rattan is critical to our discussion.

• An explicit form for Kerov’s character polynomials (with I.P. Goulden)
  Trans. Amer. Math. Soc., 359:3669-3685, 2007.

  Abstract: Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, Śniady has proved Biane’s conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an explicit expression for all terms in Kerov’s character polynomials. Our method is through Lagrange inversion.

• Permutation factorizations and prime parking functions
  Ann. Comb., 10(2):237-254, Sept. 2006.

  Abstract: Permutation factorizations and parking functions have some parallel properties. Kim and Seo exploited these parallel properties to count the number of ordered, minimal factorizations of permutations of cycle type (n) and (1, n-1). In this paper, we use parking functions, new tree enumerations and other necessary tools, to extend the techniques of Kim and Seo to permutations with cycle type (2, n-2) and (3, n-3).

• Goldbach’s conjecture in ℤ [x]. (with C. Stewart)
  C. R. Math. Acad. Sci. Soc. R. Can., 20(3):83-85, 1998.
  

  Abstract: In this paper, we prove that given any polynomial F(X) ∈ Z[X], there exist two irreducible polynomials P(X), Q(X) ∈ Z[X] such that F = P + Q. This is the analogue for Z[X] of the famous Goldbach conjecture for integers. The proof uses a combination of the Euclidean algorithm and the Eisenstein irreducibility criterion. Using the Eisenstein criterion, we also prove that there exist infinitely many polynomials F(X) ∈ Z[X] such that F²(X) + 1 is irreducible.

Theses

• Character Polynomials and Lagrange Inversion
  PhD. Thesis, University of Waterloo, 2005

  Abstract: In this thesis, we investigate two expressions for symmetric group characters: Kerov’s universal character polynomials and Stanley’s character polynomials. We give a new explicit form for Kerov’s polynomials, which exactly evaluate the characters of the symmetric group scaled by degree and a constant. We use this explicit expression to obtain specific information about Kerov’s polynomials, including partial answers to positivity questions. We then use the expression obtained Kerov’s polynomials to obtain results about Stanley’s character polynomials.

• Parking Functions and Related Combinatorial Structures
  Master’s Thesis, University of Waterloo, 2001

  Abstract: The central topic of this thesis is parking functions. A parking function is a sequence of positive integers (a₁, a₂, …, a _n) such that its non-decreasing rearrangement (b₁, b₂, …,b_n) satisfies b_i ≤ i. We give a survey of some of the current literature concerning these sequences and focus on their interaction with other combinatorial objects; namely noncrossing partitions, hyperplane arrangements and tree inversions. In the final chapter, we discuss generalizations of both parking functions and the above structures.