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MATH314301 Mathematics

创建日期：2024-04-12 16:33:07

所属分类：数学mathematics

标签： MATH314301

Mathematics

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Module Code: MATH314301

Take home exam for Combinatorics School of Mathematics

Exam information: There are 4 pages to this exam. Answer all questions.

All questions are worth equal marks. This is an open book exam. Always explain your answers,

but keep your explanation brief. For each question your final answer must be consistent with

the calculation or explana- tion. Otherwise you will get no marks for the final answer, even if it is correct.

The work that you submit must be your own work. You must not ask others to help you to complete your

assessments. If we have reason to believe that work you have submitted does not represent your own

understanding, we will take this very seriously, investigating it and following the University’s established

procedures relating to academic malpractice. You must hand in your work using Gradescope.

Turn the page over Module Code: MATH314301 1. (a) How many ways are there to distribute

4 indistinguishable apples and 6 indistin- guishable bananas into 4 distinct lunch boxes in such a way that each lunch

box contains at least two pieces of fruit? Hint. This somewhat similar to Question 4 of Workshop 1 (but note that,

unlike the books, the bananas are indistinguishable). You should first list the types of distributions of the apples.

These are the partitions of 4. For each type you deter- mine then how many ways there are to distribute the apples and then how many ways there are to distribute the bananas. (b) Let O be a set of k (distinct) objects and let B be a set of n (distinct) boxes. Show by giving a 1-1 correspondence that the number of ways of distributing the objects from O into the boxes from B is the same as the number of functions from O to B. Express this number in terms of k and n. (c) Lemma 1.23 from the notes states: The number of solutions to e1 + · · ·+ en = k where the ei are nonnegative integers is ( n+k−1 n−1 ) = ( n+k−1 k ) . Give an alternative proof to the proof from the notes as follows. Step 1. Show that the solutions (e1, . . . , en) above are in 1-1 correspondence with the solutions to f1 + · · ·+ fn = n+ k, where the fi are integers ≥ 1. Step 2. Show that the solutions (f1, . . . , fn) from Step 1 are in 1-1 correspondence with the (n− 1)-element subsets of {1, 2, . . . , n+ k − 1}. Step 3. Show that Lemma 1.23 follows from Step 1 and Step 2. Hint. For Step 2 use the map (f1, . . . , fn) 7→ {s1, . . . , sn−1} where the si are given by si = f1 + f2 + · · ·+ fi. 2. (a) Explain in your own words why the coefficient of xn in the Euler product ∞∏ k=1 1 1− xk = ∞∏ k=1 (1 + xk + x2k + x3k + · · · ) is equal to the number of partitions of n. (b) Explain which partitions of n are counted by the coefficient of xn in 1 1− x ∗ (1 + x 2 + x4) ∗ 1 1− x3 . (c) Explain what the Stirling number S(7, 5) means in terms of partitions of a set and what it means in terms of surjective functions. (d) Compute the Stirling number S(7, 5) and explain your answer. Note. There are at least three different possible methods, one reasoning with partitions directly, and others using results from the notes. Page 2 of 4 Turn the page over Module Code: MATH314301 3. (a) Consider the following problem. We are given a set O of objects and a set C of containers and we want to distribute all objects from O into the containers from C. However, not any container can contain any object: for each object y ∈ O there is a set Cy ⊆ C of containers in which y can be put. Furthermore, we want no container to contain more than r objects, where r is an integer ≥ 1. i. In Workshop 4 Question 1(b) the following m-version of Hall’s Theorem was stated. Let A1, . . . , An be finite subsets of a set X and let m be an integer ≥ 1. Then (A1, . . . , An) has a system of representatives in which each element occurs at most m times if and only if m|⋃i∈J Ai| ≥ |J | for all J ⊆ {1, . . . , n}. Use this result to deduce a condition in terms of O, the Cy and r only such that the objects from O can be distributed into the containers from C with the given restrictions if and only if the condition holds. Explain carefully what each of the quantities X,Aj,m, n in the theorem correspond to in this problem. ii. The m-version of Proposition 6.5 from the notes is obtained by replacing k by m ∗ k in condition (b). Deduce from this result a condition in terms of O, the Cy and r only such that the objects from O can be distributed into the containers from C with the given restrictions if the condition holds. Show that this condition is satisfied for r = 2 in an explicit example of your own devising with 2 ≤ |C| < |O| ≤ 4. (b) i. Illustrate the application of Proposition 6.5 in the proof of Proposition 7.4 in the notes with an explicit example where n = 3 or 4. ii. A permutation pi of {1, . . . , n} is called cyclic if, for some m ≥ 1 and an m-subset S = {a1, . . . , am} of {1, . . . , n}, we have pi(ai) = ai+1 for i ∈ {1, . . . ,m − 1}, pi(am) = a1 and pi(x) = x for x /∈ S. Now take n = 4. Let a, b, c, d ∈ {1, 2, 3, 4} be distinct. Explain how many Latin squares of order 4 there are whose first row is (a, b, c, d) and whose second row is obtained from the first by applying a cyclic permutation. Hint. Note that in our situation we must have m = n = 4 for the cyclic permutation (Why?). Page 3 of 4 Turn the page over Module Code: MATH314301 4. (a) Let X = {1, . . . , 7}. i.

What is the maximal size of an intersecting family of subsets of X? Give one

explicit example of an intersecting family F of maximal size such that for each x ∈ X

there is an A ∈ F such that x /∈ A. Explain why your example satisfies these criteria. ii.

What is the maximal size of an Sperner family of subsets of X? Give one explicit example of a

Sperner family of maximal size (you don’t have to write it out completely). (b) A town has 7000 inhabitants.

Within each week each inhabitant visits each su- permarket in the town at most once.

During a particular week the town has had more than 21000 supermarket visits by its inhabitants and each supermarket had at most 3000 visits.

Explain why at least one inhabitant must have visited at least 4 supermarkets and why the town must have at least 8 supermarkets.

(c) Let G = (V,E) be a complete graph with a red-blue edge colouring. For v ∈ V let X(v) be the set of vertices connected to v

with a red edge and let Y (v) be the set of vertices connected to v with a blue edge. Show that, when |V | is odd,

there must be a v ∈ V with |X(v)| and |Y (v)| even. Page 4 of 4 End.

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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