Department of Mathematics

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)

Current co-organizers:

Doron Zeilberger (doronzeil {at} gmail [dot] com)

Mingjia Yang (my237 {at} math [dot] rutgers [dot] edu)

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 Mingjia Yang (my237@math...).

The study of permutation patterns began with Knuth's analysis of a certain "stack-sorting algorithm" in 1968. In his 1990 PhD. thesis, West investigated a deterministic variant of Knuth's algorithm, which we can view as a function s that defines a dynamical system on the set of permutations. He defined the fertility of a permutation to be the number of preimages of that permutation under s. We will describe a colorful method for computing the fertility of any permutation, answering a question of Bousquet-Mélou. Applications of this method allow us to reprove and generalize several known theorems and improve the best known upper bound for the number of so-called "t-stack-sortable" permutations of length n when t=3 and when t=4. The method also allows us to connect the stack-sorting map with free probability theory, many well-studied combinatorial objects, and several interesting sequences. Finally, we will consider two operators on words that extend the stack-sorting map.

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to representations of Sk. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

Conolly's sequence (A046699) is defined by the nested recurrence relation C(n)=C(n-C(n-1))+C(n-1-C(n-2)) and the initial conditions C(1)=C(2)=1. This sequence is monotone increasing with each positive integer appearing at least once, a property known in the literature as slow. Conolly's sequence and several other slow sequences generated by nested recurrences are known to have combinatorial interpretations in terms of enumerating leaves in infinite tree structures. For the Conolly sequence, the tree-based interpretation illuminates an intimate connection between the sequence and the powers of two. In fact, the Conolly sequence has an alternate, purely number-theoretic definition based on powers of two. Replacing powers of two with Fibonacci numbers in this construction yields a different slow sequence (A316628). In this talk, we will describe the three different ways of constructing the Conolly sequence (recurrence, number theory, trees) and show why they all yield the same sequence. Then we will construct this new sequence based on the Fibonacci numbers, which also is slow, has a tree-based interpretation, and satisfies a nested recurrence. If time permits, we will describe how to generalize the construction to discover many new integer sequences.

Few branches of mathematics have been more influential in the social sciences than game theory. In recent years, it has become an essential tool for all social scientists studying the strategic behavior of competing individuals, firms, and countries. However, the mathematical complexity of game theory is often very intimidating for students who have only a basic understanding of mathematics. The book: