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This material will remain here until the end of August. |
The module will run in Semester 2 (January to April 2017).
The module descriptor is here. As you will see, this allows some flexibility. This year, we will be doing syllabus 1 (enumeration and formal power series). Materials for the other syllabuses are available: polynomial combinatorics, finite geometries.
The module information sheet is here.
Lectures are Mondays (odd weeks), Wednesdays, and Fridays at 12:00 in room 1A. Tutorials are Mondays at 15:00 and Tuesdays at 11:00 in room 3B. You are welcome to come to my office at any time when I am free; please see my timetable for my free hours.
This schedule for the semester is somewhat tentative!
The solutions given here do not aim to solve the problems in the most efficient way possible; but if you read them, you may learn something from them. But note that they are not model solutions to guide you in the exam!
Here are the sample exam paper and the actual paper set in 2014, the last time this syllabus was used.
The upper bound for the Ramsey number R_{2}(5,5) (the smallest number n such that n people at a party must include either 5 mutual friends or 5 mutual strangers) has been improved from 49 to 48. The paper is here. The lower bound remains at 43 (which is conjectured to be the correct value).
We showed early on in the module that the number of compositions of n (expressions of n as an ordered sum of positive integers) is 2^{n−1}, while the number of compositions of n into 1s and 2s is a Fibonacci number.
Prove that the number of compositions of n into odd numbers is also a Fibonacci number (suitably shifted).
A solution is here.
--oo--
Here is a problem for you to try. No special knowledge required.
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Let f be a polynomial which takes integer values on all the non-negative integers. Show that f takes integer values on all integers. |
Here is a solution (but try it yourself first!)
Here is Zhu Shijie's "Pascal triangle" from his book Siyuan Yujian (1303).
Peter J. Cameron
Room 317, Mathematical Institute
pjc20@st-andrews.ac.uk
19 May 2017