COMPSCI 575/MATH 513: Combinatorics and Graph Theory

Practice Second Midterm Exam, Fall 2016

David Mix Barrington

11 November 2016

Directions:

• Answer the problems on the exam pages.
• There are four problems (some with multiple parts) for 100 total points. Estimated scale is A = 93, C = 63.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• Numerical answers may be given as expressions involving arithmetic operations (including exponentiation and factorial) or the functions P(n, k) and C(n, k). In general a summation is a complete answer to a counting problem, but a recurrence or a generating function is not.

  Q1: 20 points
  Q2: 40 points
  Q3: 20 points
  Q4: 20 points
Total: 100 points

• Question 1 (20): Identify each of these statements as true or false. No explanation is needed or wanted, and there is no partial credit or penalty for guessing (2 points each).
  • (a) There are P(n, k)² ways to place k identical noncapturing rooks on an n by n chessboard.

  • (b) There are P(n, k)C(n, k)² ways to place k identical noncapturing rooks on an n by n chessboard.

  • (c) There are k!C(n, k)² ways to place k identical noncapturing rooks on an n by n chessboard.

  • (d) The number of ways to distribute r identical treats to r identical dogs is C(n+r-1, n-1).

  • (e) There are exactly fifteen ways to distribute seven identical treats to seven identical dogs, where we don't insist that each dog gets a treat.

  • (f) There is no self-conjugate partition of 7.

  • (g) If A, B, and C are any three sets, |A ∩ B ∩ C| = |A| + |B| + |C| - |A ∪ B| - |A ∪ C| - |B ∪ C| + |A ∪ B ∪ C|.

  • (h) Any recurrence of the form a_n = ca_n-1 + d, where c and d are integers, has a solution of the form a_n = dn + e, where e is an integer depending on a₀.

  • (i) If f(x) and g(x) are exponential generating functions for the sequences a₀, a₁,... and b₀, b₁,... respectively, then f(x)g(x) is the exponential generating function for c₀, c₁,..., where for each n c_n is the sum for i from 0 to n of a_ib_n-i.

  • (j) For any n, the sum for k from 0 to n of (-1)^k2^n-kC(n, k) is 1.

• Question 2 (40): In this problem we have a set of eleven dog treats {t₁,..., t₁₁} and a set of four dogs {Arly, Biscuit, Cardie, Duncan}. We will investigate the distribution of these treats to the dogs.
  • (a, 10) In how many ways can the eleven treats (considered here to be identical) be distributed to the four dogs, if each dog must get at least two treats? You many use any method, but explain how you got your answer.

  • (b, 10) Suppose now we consider the treats to be distinct, but still require that each dog get at least two treats. Give an exponential generating function for the number of ways to distribute r distinct treats to four distinct dogs, and indicate how to obtain the answer for eleven treats from this function. (You need not actually determine the answer.)

  • (c, 10) Let X be the set of all distributions of eleven identical treats to the four distinct dogs. Let A be the subset of X consisting of distributions that give Arly zero or one treats, and let B, C, and D be the analogous sets for Biscuit, Cardie, and Duncan. Determine the sizes of X, A, B, C, and D, and show how the answer for part (a) may be determined from these by Inclusion/Exclusion.

  • (d, 10) Give an ordinary generating function for the number of ways to give r identical treats to the four dogs, with each dog getting at least two treats. Express your generating function as a ratio of polynomials. Indicate how the answer for part (a) may be obtained from this function (you need not actually do it).

• Question 3 (20): These questions involve the recurrence a_n = 3a_n-2 + 2a_n-3, with initial conditions a₀ = 2, a₁ = 0, and a₂ = 7.
  • (a, 5) Determine the value of a₇, by any method.

  • (b, 5) Give the characteristic polynomial of this recurrence, and factor it as a product of linear polynomials.

  • (c, 10) Determine the general solution for this recurrence, and the specific solution for these initial conditions.

• Question 4 (20): Define the function g(n) to be the sum, for i from 0 to n, of C(n - i, i). Define the function f(n) by the recurrence f(n) = f(n-1) + f(n-2), with initial conditions f(0) = f(1) = 1. (This is Tucker's form of the Fibonacci seqeunce.) Prove, by any method, that f and g are the same function.

Last modified 14 November 2016