Download sample paper examination diet school of mathematics ... and more Study notes Mathematics in PDF only on Docsity! SAMPLE PAPER EXAMINATION DIET SCHOOL OF MATHEMATICS & STATISTICS MODULE CODE: MT5821 MODULE TITLE: Advanced Combinatorics EXAM DURATION: 21 2 hours EXAM INSTRUCTIONS: Attempt ALL questions. The number in square brackets shows the maximum marks obtainable for that ques- tion or part-question. Your answers should contain the full work- ing required to justify your solutions. PERMITTED MATERIALS: Non-programmable calculator YOU MUST HAND IN THIS EXAM PAPER AT THE END OF THE EXAM. PLEASE DO NOT TURN OVER THIS EXAM PAPER UNTIL YOU ARE INSTRUCTED TO DO SO. MT5821 Sample paper, Page 1 of 4 1. (a) What is a formal power series over a commutative ring R with identity? Give the rules for addition and multiplication of formal power series. [2] (b) Show that a formal power series with constant term 1 has a multiplicative inverse, and that a formal power series with constant term 0 and coefficient of x equal to 1 has an inverse under composition. [2] (c) What is the inverse under composition of the formal power series x − x2? Give brief proof. [3] (d) Suppose that the integers a0, a1, a2, . . . satisfy the recurrence relation an = an−1 + 2an−2 for n ≥ 2. Show that the generating function satisfies∑ n≥0 anx n = c + dx 1− x− 2x2 . Given that a0 = 1 and a1 = 5, find c and d and hence find a formula for an for all n. [3] 2. (a) State Euler’s Pentagonal Numbers Theorem, and explain its relevance to evaluating p(n), the number of partitions of the natural number n. (In particular you should explain how the theorem can be used to compute these numbers efficiently.) [3] (b) How many invertible n × n matrices over a field of q elements are there? What is the probability p(n, q) that a random n× n matrix is invertible? [3] (c) Use Euler’s Pentagonal Numbers Theorem to find a convergent series expansion in 1/q for lim n→∞ p(n, q). Hence find an approximate value of the limit for q = 2, within 1/1000 of the true value. (You may express your answer as a fraction.) [4] MT5821 Sample paper, Page 2 of 4
