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.
Mailing Address:
Department of Mathematics, Statistics, and Computer Science
Marquette University
P.O. Box 1881
Milwaukee, WI 53201-1881
A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.
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.
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.
\(\mathcal{C}\)-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.
In the antecedent paper to this it was established that there is an algebraic number \(\xi\approx 2.30522\) such that while there are uncountably many growth rates of permutation classes arbitrarily close to \(\xi\), there are only countably many less than \(\xi\). Here we provide a complete characterization of the growth rates less than \(\xi\). In particular, this classification establishes that \(\xi\) is the least accumulation point from above of growth rates and that all growth rates less than or equal to \(\xi\) are achieved by finitely based classes. A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations. We conclude by refuting a suggestion of Klazar, showing that \(\xi\) is an accumulation point from above of growth rates of finitely based permutation classes.
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.