Doctor of Philosophy (Ph.D.)
Mathematics
University of Pennsylvania
1998
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Faculty
Faculty
Alexander Burstein, Ph.D.
Professor, Chair of APT committee
Department/Office
- Mathematics
School/College
- College of Arts & Sciences
Biography
Alexander Burstein, Ph.D., is a professor in the Department of Mathematics at Howard University and a member of the Graduate Faculty. He earned his Ph.D. and M.A. in Mathematics from the University of Pennsylvania and holds a B.A., cum laude, in Mathematics from the same institution. His academic career includes faculty appointments at Iowa State University, George Washington University, and the University of Rhode Island before joining Howard University.
Burstein’s research focuses on combinatorics and graph theory, with particular emphasis on enumerative and algebraic combinatorics, pattern avoidance in permutations, permutation statistics, and the Riordan group. He has authored numerous articles in leading mathematical journals and has presented his work at national and international conferences. His scholarship explores deep structural aspects of discrete mathematics and contributes to ongoing developments in the theory of combinatorial patterns.
At Howard University Burstein teaches undergraduate and graduate courses in mathematics and mentors students engaged in research. He is actively involved in curriculum development and supports graduate student training in combinatorial theory. Through his research, teaching, and service, Burstein contributes to the department’s mission of excellence in mathematical scholarship and education.
Education & Expertise
Education
Master of Arts (M.A.)
Mathematics
University of Pennsylvania
1993
Bachelor of Arts (B.A.)
Mathematics
University of Pennsylvania
1993
Academics
Academics
College Algebra I (with ALEKS educational software)
Patterns in Mathematics
Actuarial Science Seminar
Calculus I
Calculus II
Calculus III
Research
Research
Specialty
Enumerative Combinatorics, especially Permutation Patterns, Permutation Statistics, and the Riordan GroupFunding
- NSA Young Investigator Grant, 2006-2008.
- Advanced Summer Research Fellowship, Howard University, 2012.
- Permutation Patterns conference grants:
- PI, NSF, 2016.
- Co-PI, NSF and NSA, 2013, 2017, 2018, 2019-2021 (extended through 2022).
- Co-PI, NSA, 2013, 2018, 2022.
Publications and Presentations
Publications and Presentations
Distribution of sets of descent tops and descent bottoms on restricted permutations
Distribution of sets of descent tops and descent bottoms on restricted permutations
In this note, we prove some and conjecture other results regarding the distribution of descent top and descent bottom sets on some pattern-avoiding permutations. In particular, for 3-letter patterns, we show bijectively that the set of descent tops and the set of descent bottoms are jointly equidistributed on the avoiders of 3142, 3241, 4132. This conjecture and several others made in this paper have now been proved by Zhou, Zang, and Yan (2024).
On (shape-)Wilf-equivalence of certain sets of (partially ordered) patterns
On (shape-)Wilf-equivalence of certain sets of (partially ordered) patterns
We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of patterns {12345,12354} and {45123,45213} that extends the list of 10 related conjectures proved in the literature in a series of papers. To achieve our goals, we prove generalized versions of shape-Wilf-equivalence results of Backelin, West, and Xin and use a particular result on shape-Wilf-equivalence of monotone patterns. We also derive general results on shape-Wilf-equivalence of certain classes of partially ordered patterns and use their specialization (also appearing in a paper by Bloom and Elizalde) as an essential piece in proving the conjecture. Our results allow us to show (shape-)Wilf-equivalence of large classes of sets of patterns, including 11 out of 12 classes found by Bean et al. in relation to the conjecture.
Distribution of peak heights modulo k and double descents on k-Dyck paths
Distribution of peak heights modulo k and double descents on k-Dyck paths
We show that the distribution of the number of peaks at height i modulo k in k-Dyck paths of a given length is independent of i∈[0,k−1] and is the reversal of the distribution of the total number of peaks. Moreover, these statistics, together with the number of double descents, are jointly equidistributed with any of their permutations. We also generalize this result to generalized Motzkin paths and generalized ballot paths.
Permutations with exactly one copy of a monotone pattern of length k, and a generalization
Permutations with exactly one copy of a monotone pattern of length k, and a generalization
We construct an injection from the set of permutations of length n that contain exactly one copy of the decreasing pattern of length k to the set of permutations of length n+2 that avoid that pattern. We then prove that the generating function counting the former is not rational, and in the case when k is even and k≥4, it is not even algebraic. We extend our injection and our nonrationality result to a larger class of patterns.
Pseudo-involutions in the Riordan group
Pseudo-involutions in the Riordan group
We consider pseudo-involutions in the Riordan group where the generating function g for the first column of a Riordan array satisfies a palindromic or near-palindromic functional equation. For those types of equations, we find, for very little work, the pseudo-involutory companion of g and have a pseudo-involution in a k-Bell subgroup. There are only slight differences in the ordinary and exponential cases. In many cases, we also develop a general method for finding B-functions of Riordan pseudo-involutions in k-Bell subgroups, and show that these B-functions involve Chebyshev polynomials. We apply our method for many families of Riordan arrays, both new and already known. We also have some duality and reciprocity results. Since many of the examples we discuss have combinatorial significance, we conclude with a few remarks on the general framework for a combinatorial interpretation of some of the generating function results we obtain.