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# Student Research

# Student Research

Some of our students are involved in conducting original mathematics research. Research projects vary in nature and scope; they often take place during the summer, and often carry significant amounts of financial support. In the past, students have participated in research that has led to presentations at conferences, and even publications in professional journals. Below are summaries of some of the most recent projects our students have worked on.

The Search for the Cyclic Sieving Phenomenon in Plane Partitions

Research by William J. Asztalos

Abstract: The efforts of this research project are best understood in the context of the subfield of dynamical combinatorics, in which one enumerates a set of combinatorial objects by defining some action to guide the search for underlying structures. While there are many examples with varying degrees of complexity, the necklace problem, which concerns the possible unique configurations of beads in a ring up to rotational symmetry, is a well-known example. Though this sort of approach to enumeration has been around for a century or more, activity in this area has intensified in the last couple of decades. Perhaps the most startling development was the discovery of the cyclic sieving phenomenon, in which polynomial generating functions produce information about the sizes of rotational symmetry classes of objects. This technique is an extension of the “q = -1” phenomenon which classifies objects on the basis of being a fixed point or an element of a “mirrored” pair. In this study, we are on the hunt for rotational symmetries in plane partitions, with the ultimate goal of recovering the “magic” polynomial that will allows us to count the symmetry classes of these objects. The unique characteristics of plane partitions under our devised operation portend that attaining such a goal is feasible.

Exact Recovery of Prototypical Atoms through Dictionary Initialization

Research by Greg Zanotti (student) and Enrico Au-Yeung (faculty)

Abstract: In dictionary learning, a matrix comprised of signals Y is factorized into the product of two matrices: a matrix of prototypical "atoms" D, and a sparse matrix containing coefficients for atoms in D, called X. Dictionary learning finds applications in signal processing, image recognition, and a number of other fields. Many algorithms for solving the dictionary learning problem follow the alternating minimization paradigm; that is, by alternating solving for D and X. In 2014, Agarwal et al. proposed a dictionary initialization procedure that is used before this alternating minimization process. We show that there is a modification to this initialization algorithm and a corresponding data generating process under which full recovery of D is possible without a subsequent alternating minimization procedure. Our findings indicate that the costly step of alternating minimization can be bypassed, and that other data models may enjoy the same features as the one we propose.

Signal processing on graphs using Kron reduction and spline interpolation

Research by Michael Dennis and Enrico Au-Yeung (faculty)

Abstract: In applications such as image processing, the data is given in a regular pattern with a known structure, such as a grid of pixels. However, it is becoming increasingly common for large data sets to have some irregular structure. In image recognition, one of the most successful methods is wavelet analysis, also commonly known as multi-resolution analysis. Our project is to develop and explore this powerful technique in the setting where the data is not stored in the form of a rectangular table with rows and columns of pixels. While the data sets will still have a lot of structure to be exploited, we want to extend the wavelet analysis to the setting when the data structure is more like a network than a rectangular table. Networks provide a flexible generalization of the rigid structure of rectangular tables.

Power Series for Up-Down Min-Max Permutations

Calculus and combinatorics overlap, in that power series can be used to study combinatorially defined sequences. In this project, we used exponential generating functions to study a curious refinement of the Euler numbers, which count the number of “up-down” permutations of length n.

Published in: Heneghan, Fiacha and Petersen, T. Kyle, "Power series for Up-Down Min-Max Permuations," College Mathematics Journal, Vol. 45, No. 2, March 2014, p.83-89.

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Unimodality via alternating gamma vectors

Research by Charles Brittenham, Andrew Carroll (faculty), T. Kyle Petersen (faculty) , and Connor Thomas

We attempted to find combinatorial proofs of unimodality for various number sets, namely the q-analogue of n!, the q-binomial coefficients, and integer partitions with distinct parts of size at most n. We proceeded by attempting to find a sign-reversing involution on the gamma-vector expansions for each of these polynomials to show that the entries of these vectors were nonnegative, and hence that the polynomials modeled by those gamma-vectors are unimodal. While we were able to show this for the q-analogue of n!, further refining of the involutions for the remaining two number sets is needed to give a complete proof of unimodality.

Published as: *Unimodality via alternating gamma vectors*, Electron. J. Combin. **23** (2016), no. 2, Paper 2.40, 22 pp.

In this project, we studied a curious refinement of the Euler numbers, which count the number of "up-down" permutations of length

*n*. Specifically, we defined two sequences of numbers that counted up-down permutations according to whether or not the digit 1 occurred before the digit

*n*. Using combinatorial reasoning, we were able to discover generating functions for these sequences of integers. We then used the generating functions to investigate further properties of these sequences.

Published in: Heneghan, Fiacha and Sliva, Ashley (2013) "Max-Min Up-Down Permutations," DePaul Discoveries: Vol. 2: Iss. 1, Article 1.

Enumerating Alternating Permutations with One Alternating Descent

In this project we introduced a new statistic for alternating permutations, called an alternating descent. We focused on alternating permutations with one alternating descent, and were able to enumerate these permutations by decomposing them into four sets.

Published in: Wagner, Stacey (2013) "Enumerating Alternating Permutations with One Alternating Descent,"DePaul Discoveries: Vol. 2: Iss. 1, Article 2.

A Generalization of Pascal's Triangle

Combinatorics is a branch of mathematics interested in the study of finite, or countable, sets. In particular, Enumerative Combinatorics is an area interested in counting how many ways patterns are created, such as counting permutations and combinations. Brenti and Welker, authors of “The Veronese Construction for Formal Power Series and Graded Algebras,” seek an explanation for a combinatorial identity posed in their research. Using techniques practiced in this area of mathematics, we have discovered that certain numbers appearing in their identity hold properties similar to properties of the well-known binomial coefficients.

Published in: Gwetta, Eliya; Pacurar, Adrian; and Smith, Elizabeth Mai (2012) "A Generalization of Pascal’s Triangle," DePaul Discoveries: Vol. 1: Iss. 1, Article 13.