# Mathematics

## Colloquium

#### Upcoming Seminar

December 5, 4:10 PM in Physics 123
On counting upper interactions in Dyck paths
Yvan Le Borgne, Simon Fraser University & CNRS/LaBRI (Université Bordeaux)

The Catalan numbers forms the counting sequence of many classes of objects in enumerative combinatorics (see R.P. Stanley Catalan addendum). This sequence frequently results from the existence of a recursive decomposition of objects into one atomic element and two, possibly empty, smaller objects of the same class. In terms of generating functions, this decomposition turns into an algebraic equation of degree two.

A classical interpretation of the Catalan numbers are the Dyck words. A Dyck word of size $$n$$ is a word on the alphabet $$\{a,b\}$$ which contains $$n$$ occurrences of each letter and where each prefix contains at least as many occurrences of letter $$a$$ than occurrences of letter $$b$$. An upper interaction in a Dyck word is any occurrence of a factor $$b^ka^k$$ with $$k\geq 1$$. The enumeration of Dyck words according to their size and number of upper interactions creates a correlation between the two subwords of the usual recursive decomposition of Dyck words.

The aim of this talk is to explain various ways to tame this correlation, revisiting some of the classical methods to count Dyck words via a generating function. This will involve the kernel method, solving some $$q$$-algebraic equations, the theory of heaps and some formal regular/algebraic languages.

Based on this and additional recent progress by the speaker.

Most Thursday afternoons during the academic year, the Reed College Department of Mathematics hosts a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels. Refreshments are served before the talks. For more information, please email davidp at reed.edu.

### 2013-14 Schedule

#### Fall

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Sept 5 Meeting with majors. No talk this week. Student PresentationsMaddie Brandt, Joseph Rennie, Qiaoyu Yang, and Kuai YuMaddie Brandt:  Packings of Four Equal Circles on Flat Tori Joseph Rennie:  Algorithms for Determining Zero Set Topologies Qiaoyu Yang and Kuai Yu:  Parking functions and tree inversions Maximally Intransitive DiceJoe Buhler, CCR/Reed CollegeIt's been known for a long time that there are three dice $$A$$, $$B$$, $$C$$ such that $$A > B > C > A$$ where "$$A> B$$" means that $$A$$ beats $$B$$ in the sense that the probability that a roll of $$A$$ is higher than a roll of $$B$$ is strictly bigger than 1/2. In fact there are dice such that $$A>B>C>A$$ and yet \(A^{[2]}

#### Spring

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Jan 30 Greg Smith, Department of Mathematics and Statistics, Queens University Spring break Melody Chan, Department of Mathematics, Harvard University