## RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

and the

### Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Founded 2003 by Drew Sills and Doron Zeilberger.

Former co-organizers: Drew Sills (2003-2007), Moa ApaGodu (2005-2006), Lara Pudwell (2006-2008), Andrew Baxter (2008-2011), Brian Nakamura (2011-2013), Edinah Gnang (2011-2013), Matthew Russell (2013-2016), Nathan Fox (2016-2017), Bryan Ek (2017-2018), Mingjia Yang (2018-2020), Yonah Biers-Ariel (2018-2020)

Current co-organizers:
Doron Zeilberger (doronzeil {at} gmail [dot] com)
Robert Dougherty-Bliss (robert {dot} w {dot} bliss {at} gmail [dot] com)

Archive of Previous Speakers and Talks You can find links to videos of some of these talks as well. Currently, our videos are being posted to our Vimeo page. Previously, we had videos posted on our YouTube page.

If you would like to be added to the weekly mailing list, email Robert Dougherty-Bliss (robert {dot} w {dot} bliss {at} gmail [dot] com)

# Fall 2021 Semester

Date: Sept. 23 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Noga Alon, Princeton University

Title: Coloring subsets with r-wise intersecting color classes

Abstract: What is the minimum number of colors required in a coloring of all k-subsets of an n-set so that every color class is r-wise intersecting? We suggest a conjectured answer for all r, k and n, note that for r=2 this is Kneser's conjecture proved by Lovasz, and prove the conjecture for any r which is either a prime or a power of 2.

Date: Sept. 30 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Doron Zeilberger, Rutgers University

Title: An Experimental (yet fully rigorous!) Study of a certain "Measure Of Disarray" that 12-year Noga Alon Proved was always Even

Abstract: In a beautiful new "cofee table book", "Do not Erase", by the very talented artistic photographer Jessica Wynne, there are pictures of more than one hundred blackboards by a very diverse set of mathematicians. One of them is Noga Alon's blackboard. Each blackboard photo is accompanied by a short essay by the creator of the blackboard, where they often describe how they decided to become mathematicians. According to Noga Alon, the "epiphany" occurred when he was 12 years old, when he settled a heated argument in a "Eurovision watching party" that his parents threw, where he conclusively proved that a certain "measure of disarray" must always be even, in particular causing one of the guests, "a grown-up engineer", to concede that he was wrong in claiming that it was a coincidence that the scores turned out to be all even. According to Noga, the fact that a mere 12-year-old can win an argument against a grown-up (especially one who is an engineer) shows the objectivity of mathematical truth, and reinforced his decision to become a mathematician.

While I do agree that mathematical knowledge is more objective than most other kinds, it is not as objective as it seems. But the point of the present talk is to investigate, in a purely empirical (yet fully rigorous!) way, that same measure of disarray, that turned out to be called "Spearman's footrule", going far beyond just proving that it is always even, considerably extending a 1977 paper by Persi Diaconis and Ron Graham, debunking yet-another myth: that mathematics is always deductive.
Joint work with Shalosh B. Ekhad.

Date: Oct. 7, 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Yochay Jerby, Holon Institute of Technology, Israel

Title: A dynamic approach for the zeros of the Riemann zeta function - Collision and repulsion

Abstract: he Riemann hypothesis is a question regarding the solutions of the transcendental equation ζ(s)=0, that is the zeros of the Riemann zeta function. The starting point of our talk is a remarkable connection between the zeros of zeta and the solutions of the much simpler equation χ(s)+1=0, whose non-trivial solutions could be completely described in closed form and all lie on the critical line. We will explain that the two equations are related by a sequence of functions ζN(s) called $N$-th sections of zeta. For N=1 one has ζ1(s)=1/2(1+χ(s)) while for N=[Im(s)/2] the section ζN(s) is approximately ζ(s), up to a small error. Studying the dynamical change of the zeros of ζN(s) with respect to N we will show that zeros can go off the critical line only if a process of collision occurs between two consecutive zeros at a certain stage. We will also suggest a method of re-arranging the dynamics so that collisions could be avoided altogether, that is, in a way expected to keep the zeros on the critical line for any N, including the final stage where they essentially coincide with the zeros of zeta. If time permits we will also discuss how the suggested viewpoint relates to various classical topics such as: Gram's law, the Davenport-Heilbronn function and the Montgomery pair correlation conjecture.

Date: Oct. 14 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Stoyan Dimitrov, University of Illinois Chicago

Title: Moments of permutation statistics and central limit theorems

Abstract: We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns which uses a lemma of Burstein and Hasto. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns tbd

Date: Oct. 21 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Orli Herscovici, Georgia Tech

Title: Combinatorics behind the degenerate Eulerian numbers

Abstract: Works of Carlitz gave an inspiration to many researchers to develop different generalizations of the Eulerian polynomials and numbers. Many of those generalizations have a pure analytical character. It is known that the classical Eulerian numbers and some of their generalizations are connected to combinatorial statistics on permutations. However, the degenerate Eulerian numbers had no combinatorial interpretation since their introduction by Carlitz in 1979. In this talk we consider a combinatorial model that generalizes the standard definition of permutations and show its relation to the degenerate Eulerian numbers.

Date: Oct. 28 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Eugene Zima, Wilfrid Laurier University

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Date: Nov. 4 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

Speaker: Eric Rowland, Hofstra University

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Date: Nov. 11 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

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Date: Nov. 18 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

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Date: Dec. 2 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

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Date: Dec. 9 , 2021, 5:00pm (Eastern Time) Zoom Link [password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

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