MAT 4199C/5107 Second Homework Assignment Due 13 Feb by 1:00pm

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1 Marks: LAST NAME: First name: Student number: MAT 4199C/5107 Second Homework Assignment Due 13 Feb by 1:00pm Instructions: The question labelled (U) is for undergraduate students only, while question labelled [G] is for graduate students only. The remaining questions are for both groups. MAT 4199C: Your full mark is 25. MAT 5107: Your full mark is 30. Print out this document and staple the pages. You may write on both sides of the paper or insert additional pages if necessary. Submit a finished, presentable product. Drafts and illegible papers will not be marked. Show all relevant work to receive full credit. Submit the assignment in class. Late assignments will not be accepted. Important note on academic integrity: Students are permitted, and indeed encouraged, to discuss homework problems with others, but are not permitted to help each other write the final solutions (unless the assignment is explicitly announced as a group assignment). Once you understand a solution, you must write it out entirely by yourself. For each question, any help from other people must be clearly acknowledged, as well as any sources used (e.g. textbooks, websites, videos), if that source contains a solution to a very similar question, or a new method or idea that you used that was not in the course materials. Failure to follow these rules constitutes plagiarism (academic fraud). Note that if one student copies from the other, both students have committed academic fraud. If I believe plagiarism has occurred, the students will receive: a mark of 0 for the current assignment if this is the first offence; a mark of 0 for the whole assignment component of the course if this is the second offence. Students are advised to carefully examine the University Guidelines on Academic Integrity see as well as the Course Policy on Plagiarism see Please sign below to confirm that you have read, understand, and will follow these guidelines. Only signed papers will have the mark recorded. Student s signature:

2 (Q1) Let A be the class of words over [4] that do not contain the subword 12. The size of a word is its length (number of symbols). [8pts] (a) Give a symbolic recursive construction for A. (b) Use (a) to find an explicit, closed-form OGF A(z) for A. Hint: it should be a rational function. (c) Find A n using a partial fractions decomposition. Your result should be in terms of the roots of the denominator in A(z) and the coefficients in the partial fractions decomposition, but you need not find explicit values for these parameters. (d) Determine the asymptotic behaviour of A n (as n ). (e) Use (b) to find a recurrence relation for (A n ) n 0. Give an appropriate number of initial conditions.

3 (Q2) Let d be a fixed positive integer. Consider the class D of unlabelled rooted plane trees in which each vertex has the number of children divisible by d. The size of a tree in D is defined to be the number of its vertices. [7pts] (a) Give a symbolic recursive construction for D. (b) Use (a) to find a functional equation for its OGF D(z). (c) Use the Lagrange Inversion Formula to find a simple expression for the number D n of trees in D with n vertices. (d) Verify your result in (c) for d = 1 by comparing with a result from class.

4 (Q3) Let N be the class of integer partitions whose parts are not multiples of 3. Let T be the class of integer partitions such that each integer appears at most twice. [6pts] (a) Determine the OGF for N. (b) Determine the OGF for T. (c) Show that N(z) = T (z), and conclude that N n = T n for all n.

5 (Q4)-[U] Let r be a fixed positive integer, and C be the class of integer compositions with all parts from the set [r]. [4pts] (a) Give a symbolic construction for C. (b) Use (a) to find a closed-form expression for the OGF C(z). (c) Verify your result in (b) for r = 1.

6 (Q4)-[G] Fix a prime power q, and let I be the class of irreducible monic polynomials over F q (of degree at least 1). Let D be the class of monic polynomials σ over F q such that each irreducible factor in σ occurs to a power strictly less than d, for a fixed integer d 2. Furthermore, let N be the class of monic polynomials over F q with no repeated irreducible factors. Let I(z), D(z), and N(z) be the OGFs for I, D, and N, respectively. [9pts] (a) Give a symbolic construction for D in terms of I. (b) Find an expression for D(z) in terms of the coefficients I n of I(z). (c) Give an exponential form of D(z), that is, write D(z) in the form exp(...) such that... contains I(...) but no coefficients I n. Hint: See how we derived the OGFs for SET and M SET, and their alternative (exponential) forms. (d) Give a symbolic construction for N in terms of I. (e) Give an exponential form of N(z) (in terms of I(...)). (f) Verify that your result in (e) is equivalent to your result in (c) (for an appropriate value of d).