COMPSCI 575/MATH 513: Combinatorics and Graph Theory

Second Midterm Exam, Fall 2016

David Mix Barrington

17 November 2016

Directions:

• Answer the problems on the exam pages.
• There are four problems (some with multiple parts) for 100 total points. Actual scale was A = 90, C = 54.
• 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: 25 points
  Q4: 15 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) The rook polynomial for a 2 × 3 rectangle is 1 + 6x + 9x².

  • (b) The rook polynomial for the board made by five squares, arranged in a plus sign, is 1 + 5x + 4x² + x³.

  • (c) If |A| = 5 and |B| is 4, then the number of onto functions from A to B is exactly 4×C(5, 2)×3!.

  • (d) The number of partitions of 7 into three non-empty parts is equal to the number of partitions of 7 into parts whose size is at most three.

  • (e) The recurrence a_n = a_n-1 + a_n-3 + n³ - 6 has a unique solution with the initial conditions a₀ = 0 and a₁ = 1.

  • (f) If f(x) = 1 + a₁x + a₂x² + a₃x³ + ... is any power series with integer coefficients, then there exists a power series g(x) with integer coefficients such that g(x)(1 + x + x² = f(x).

  • (g) The number of ways to distribute r distinct treats to n identical dogs is n^r.

  • (h) Consider paths in a Manhattan-like grid where edges go only in the positive x or positive y direction. The number of paths from (0, 0) to (n, n) is equal to the sum, for i from 1 to n, of C(n, i)C(n, n-i).

  • (i) For any n, the number of ternary sequennces of length n with at least one occurrence of each letter is 3ⁿ - 3×2ⁿ + 3×1ⁿ - 1.

  • (j) The words BASINGSTOKE and BELCHERTOWN each have exactly P(11, 2) = 11!/9! anagrams. (An anagram of a word is any word (whether a valid English word or not) containing the same multiset of letters.)

• Question 2 (40): In this problem we distribute a set of 13 treats {t₁,..., t₁₃} to the set of dogs {c, d, s} (Cardie, Duncan, and Scout). We will insist that Cardie and Duncan each get at least three treats, and that Scout gets an even number of treats (possibly 0).
  • (a, 10) Determine, by any method, the number of ways to distribute 13 identical treats to the three distinct dogs, with the requirements above.

  • (b, 10) Find an ordinary generating function whose x^r coefficient is the number of ways to distribute r identical treats to the three distinct dogs, while following the requirements stated above. Determine the x¹³ coefficient of this generating function.

  • (c, 10) Find an exponential generating function for the number of ways to distribute r distinct treats to the three distinct dogs, meeting the requirements above. Express this function using exponential functions and polynomial. You need not evaluate its x¹³ coefficient, but give a brief idea of how you would do so.

  • (d, 10) Express the solution to part (a) using Inclusion/Exclusion, where N is the total number of ways to distribute 13 identical treats to three distinct dogs, and A₁, A₂, and A₃ are the sets of distributions that violate each of the three conditions abobve, respectively. Find the sizes of N, A₁, A₂, and A₃ -- you need not find the sizes of the various intersections but you should express your final answer in terms of them.

• Question 3 (25): This problem deals with two recurrences. The first has rule b_n = 2b_n-1 + 3b_n-2, with initial conditions b₀ = 1 and b₁ = -1. The second has rule a_n = 2a_n-1 + 3a_n-2 + 16n - 36, with initial conditions a₀ = -3 and a₁ = 3.
  • (a, 5) Find the values of a₁ and b_i for all i with 0 ≤ i ≤ 5.

  • (b, 5) Find a general solution for recurrences with the rule of {b_n}, and find the specific solution with the given initial conditions.

  • (c, 5) Find a particular solution for the inhomogeneous part of the recurrence {a_n}.

  • (d, 10) Find a general solution for recurrences with the rule of {a_n}, and find the specifc solution with the given initial conditions.

• Question 4 (15): For any integer n with n ≥ 0, let f(n) = ∑ⁿ_i=0∑ⁱ_j=0 (-2)^i-jC(n, i)C(i, j). Prove that for all positive n, f(n) = 0. (On the exam as given, it asked to prove this for all non-negative n, but it happens to be false for n = 0.)

Last modified 30 November 2016