Research

My research centres on algebraic and enumerative combinatorics. I have worked on topics such as combinatorial representation theory, factorizations problems in the symmetric group, lattice paths and integer partitions.

Publications

• On the average number of cycles in conjugacy class products (with J. Campion Loth)
• Abstract: We show that for the product of two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types α and β, the average number of cycles in their product is between H_n−3 and H_n+1, where H_n is the harmonic number.

• A comparison of integer partitions based on smallest part (with D. Binner)
• Electon. J. Combin. Vol. 31(1), 2024.
• Abstract: For positive integers n,L and s, consider the following two sets that both contain partitions of n with the difference between the largest and smallest parts bounded by L: the first set contains partitions with smallest part s, while the second set contains partitions with smallest part at least s+1. Let G_L,s(q) be the generating series whose coefficient of qn is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu (2019). Previous results concentrated on the nonnegativity of G_L,s(q) in the cases s=1 and s=2. In the present paper, we show the eventual positivity of G_L,s(q) for general s and also find a precise nonnegativity result for the case s=3.

• k-Factorizations of the full cycle and generalized Mahonian statistics on k-forests (arxiv version) (with J. Irving)
• Adv. Appl. Math. 151, 2023 (journal version).
• Abstract: We develop direct bijections between the set F_n^k of minimal factorizations of the long cycle (0,1,...,kn) into (k+1)-cycle factors and the set of rooted labelled forests on vertices with edges coloured with {0,1,...,k-1} that map natural statistics on the former to generalized Mahonian statistics on the latter. In particular, we examine the generalized major index on forests and show that it has a simple and natural interpretation in the context of factorizations. Our results extend our own results (2021), which treated the case k=1 through a different approach, and provide a bijective proof of the equidistribution observed by Yan (1997) between displacement of k-parking functions and generalized inversions of k-forests.

• Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson (with A. Foley, A. Morales and K. Yeats)
• Algebr. Comb. 5(6):1205-1226, 2022.
• Abstract: In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {\em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collaborative work.

• On conjectures concerning the smallest and missing parts of integer partitions (arxiv version) (with D. Binner)
• Ann. Comb. 25(3):697-728, 2021 (journal version).
• Abstract: For positive integers s and L≥3, Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval {s,…,L+s}. Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases s=1 and s=2. In the present article, we prove the conjecture for general s by proving a stronger theorem. We also prove other related conjectures found in the same paper.

• Trees, parking functions and factorizations of full cycles (arxiv version) (with J. Irving)
• European J. Combin. 93, 2021 (journal version).
• Abstract: Parking functions of length n are well known to be in correspondence with both labelled trees on n+1 vertices and factorizations of the full cycle (0, 1, ... n) into transpositions. In fact, these correspondences can be refined: Kreweras equated the area enumerator of parking functions with the inversion enumerator of labelled trees, while an elegant bijection of Stanley maps the area of parking functions to a natural statistic on factorizations of . We extend these relationships in two principal ways. First, we introduce a bivariate refinement of the inversion enumerator of trees and show that it matches a similarly refined enumerator for factorizations. Secondly, we characterize all full cycles such that Stanley’s function remains a bijection when the canonical cycle is replaced by . We also exhibit a connection between our refined inversion enumerator and Haglund’s bounce statistic on parking functions.

• On products of long cycles: short cycle dependence and separation probabilities (with V. Féray)
• J. Algebraic. Combin. 42(1):183-224, 2015.
• Abstract: We present various results on multiplying cycles in the symmetric group. One result is a generalisation of the following theorem of Boccara (Discret Math 29:105–134, 1980): the number of ways of writing an odd permutation in the symmetric group on symbols as a product of an (n)-cycle and an (n-1)-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths. Another application is related to the recent notion of separation probabilities for permutations introduced by Bernardi et al. (Comb Probab Comput 23:201–222, 2014).

• Enumeration of noncrossing pairings on bit strings (with T. Kemp, K. Mahlburg and C. Smyth)
• J. Combin. Theory Ser. A 118(1):129-151, 2011.
• 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.

• 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 C√n rows and columns, for some constant C) in the limit as n tends to infinity. We find an asymptotic bound for characters in terms of a parameter D, whicn only depends on C. 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) in Z[X], there exist two irreducible polynomials P(X), Q(X) in 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) in Z[X] such that F²(X) + 1 is irreducible.