I am a mathematician at Marquette University. My research is focused on the development and application of symbolic and analytic methods in enumerative combinatorics. These methods incorporate techniques from a variety of areas, including analytic combinatorics, computer algebra, experimental and computational mathematics, and statistical mechanics.
\(\mathcal{C}\)-machines are a class of sorting machines that naturally generalize stacks and queues. A \(\mathcal{C}\)-machine is a container that is allowed to hold permutations from the class \(\mathcal{C}\) into which entries can be pushed and out of which entries may be popped. With this notation, a stack is the \(\operatorname{Av}(12)\)-machine. This structural description allows us to find many terms in the counting sequences of three permutation classes of interest, but despite these numerous initial terms we are unable to find the exact or asymptotic behavior of their generating functions. In this talk I'll describe what we do know, what we don't know, and what experimental methods tell us we might one day know.
There is a well-known upper bound on the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using staircases of permutation classes, we provide sufficient conditions for this upper bound to be achieved. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques can be used to reprove a result of Bóna.
We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation.
PermPy is a python library for the manipulation of permutations and permutation classes. It is useful for experimentation and forming conjectures, and has played a role in about a dozen publications in the field of permutation patterns.
In enumerative combinatorics, it is quite common to have in hand a number of known initial terms of a combinatorial sequence whose behavior you'd like to study. In this talk we'll describe the Method of Differential Approximants, a technique that can be used to shed some light on the nature of a sequence using only some known initial terms. While this method is, on the face of it, experimental, it often leads the way to rigorous proofs. We'll exhibit the usefulness of this method through a variety of combinatorial topics, including chord diagrams, permutation classes, and inversion sequences.
S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic \(m_1 \times \cdots \times m_d\) tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case \(m_1 = \cdots = m_d\). We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effect observed by Ekhad and Zeilberger for a particular case in fact occurs more generally.