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Proudly Serving Math Community at the Claremont Colleges Since 2007

Copyright © 2011 Claremont Center for the Mathematical Sciences

Sam Miner (Pomona College)

A permutation pattern is a sub-permutation within a longer permutation. If a long permutation does not contain a specific shorter pattern, we say it avoids the shorter pattern. In recent years, avoidance of different patterns has been systematically investigated, and many questions about the subject have been answered. In this talk, we will discuss historical results, and recent progress on the enumeration and asymptotic behavior of certain pattern-avoiding classes.

Millikan 2099, Pomona College

Mark C. Wilson (University of Auckland, New Zealand)

Lattice paths are a classical topic in combinatorics, with many applications. I report on joint work with Stephen Melczer (PhD student, Lyon/Waterloo). We consider the computation of $f_n(S)$, the number of $n$-step nearest-neighbor walks on the two dimensional non-negative integer lattice with a finite set $S$ of allowable steps. Up to isomorphism there are 79 models to consider, and previous work has shown that 23 of these satisfy a well-behaved recursion for $f_n(S)$. In 2009, Bostan and Kauers guessed asymptotics of $f_n(S)$ in these 23 cases. We provide, for the first time, a complete rigorous verification of these guesses. Our technique is to express 19 of the 23 GFs as diagonals of trivariate rational functions, and apply recently derived general methods of analytic combinatorics in several variables, as described in my 2013 book with Robin Pemantle. This approach also shows a direct link between combinatorial properties of the models and features of its asymptotics. In addition, we give using the same methodology expressions for the number of walks returning to the $x$-axis, the $y$-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech. I will attempt to cover all required background and will present many examples. If time permits I will explain how similar but less comprehensive results can be obtained in arbitrary dimension.

Millikan 2099, Pomona College

Bianca Thompson (HMC)

Using essentially only algebra, we give a proof that a cubic rational function over the complex numbers with real critical points is equivalent to a real rational function. We also show that the natural generalization to the p-adic rationals and number fields fails.

Millikan 2099 (Pomona College)

Ahmad Shaar (HMC)

In this talk, we will discuss the relationship between pseudo-random sequence circuit generators (and sequence set generators) and the algebra of Galois fields. We will start with a brief review of Galois fields and their basic properties, as well as different types of multiple access techniques. We will then talk about binary sequences (maximal length sequences, Gold sequence sets, and Kasami sequence sets) and multilevel sequences (one-coincidence sequence sets).

Millikan 2099 (Pomona College)

Alan Haynes (University of Houston)

The Steinhaus theorem, known colloquially as the 3-distance theorem, states that for any positive integer N and for any real number x, the collection of points nx modulo 1, with 0 < n < N, partitions R/Z into component intervals which each have one of at most 3 possible distinct lengths. Many authors have explored higher dimensional generalizations of this theorem. In this talk we will survey some of their results, and we will explore a two-dimensional version of the problem, which turns out to be closely related to the Littlewood conjecture. We will explain how tools from homogeneous dynamics can be employed to obtain new results about a problem of Erdos and Geelen and Simpson, proving the existence of parameters for which the number of distinct gaps in a higher dimensional version of the Steinhaus problem is unbounded.

Millikan 2099 (Pomona College)

Mohamed Omar (HMC)

We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $\P(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive integers, we define $\P(S;n)=\{\pi\in\mathfrak{S}_n:\P(\pi)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| \P(S;n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all nonnegative. In this talk we introduce a new recursive formula for $|\P(S;n)|$ without alternating sums and we use this recursion to prove that their conjecture is true. Moreover, we develop a generalization of peak polynomials to graphs that is consistent with the $S_n$ case and explore positivity therein.

Millikan 2099 (Pomona College)

Mark Huber (CMC)

The Fibonacci sequence has long been studied for its wonderful properties, including the fact that the ratio of successive terms approaches the Golden Ratio. In order to understand why this happens from a probabilistic perspective, I'll build a computer experiment over n different {0,1} random variables where the probability of the outcome being true is the n-th Fibonacci number divided by 2^n. By extending the probability distribution to infinite graphs, it becomes possible to find this limit of successive terms for large n as well as rederive the classic formula for the n-th Fibonacci number in terms of the Golden Ratio.

Millikan 2099, Pomona College

Shahriar Shahriari (Pomona College)

Let k be a fixed small positive integer and let A be a matrix. Can you find a basis for the nullspace of A consisting of vectors whose entries are from the set {±1, ±2, . . . , ±(k − 1)}? We conjecture that the answer is yes, if A is the (0, 1)–incidence matrix of a finite graph and if k = 5. If true, this would strengthen a celebrated conjecture of Tutte from the 1950s. In this talk, we discuss the conjecture, and, present positive results for a variety of graphs including the complete graphs. Joint work with Saieed Akbari and Amir Hossein Ghodrati.

Millikan 2099, Pomona College

Daqing Wan (UC Irvine)

The subset sum problem over finite fields is a well known NP-complete problem, with a wide range of applications in coding theory, cryptography and computer science. In this talk, we will review this problem from both complexity and algorithm points of view. Then we propose a new algebraic variant, where many new interesting questions arise and some of them could be studied by methods from combinatorics and number theory.

Millikan 2099, Pomona College

Jozef Przytycki (George Washington University)

We describe in this talk a new invariant of rooted trees and rooted graphs. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers of is that we deal here with an elementary, interesting, new mathematics, and after this talk listeners can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology.

Millikan 2099, Pomona College