MAT344H1S Introduction to Combinatorics Midterm assignment. Due March 1, 10:00 pm 1. In a string, distance between two symbols is the number of symbols between them plus one. For example, in a string xyyx, y is at distance 1 from another y, and x is at distance 3 from another x. Find the number of strings of length n using at most k distinct symbols, such that the minimal distance between any two identical symbols is k. 2. For a given positive integer k, a quadratic polynomial with bounded integer coefficients has the form P(x) = ax2+bx+ c, where a, b, c are integers in the range [−k, k]. There are 2k(2k + 1)2 of these — note that a cannot be 0. Show that there are k(3k + 1) of these polynomials which have 1 as a root, that is, with P(1) = 0. 3. Show that, if the minimal number of edges that can be removed from an Eulerian graphG so that it stays Eulerian is k,G has a subgraph isomorphic to the cyclic graph Ck. 4. Let f(n) denote the number of ways to cover a 2 × n rectangle with n rectangles of size 2× 1 that are blue or red. Give a combinatorial proof that f(2k+ 1) = 2f(k)2 + 8f(k)f(k− 1). 5. In the group stage of a tournament, 24 teams are separated into 6 groups of 4. Every team plays each other team in their group twice, playing 6 games total. Games may end in a tie. Show that there will be two teams with the same win-loss-tie record. 6. If a 6× 6matrix contains 19 zeros, show that there are 4 zeros which are all in distinct rows and columns. 7. n people participate in a costumed party with n different costumes. Each person likes exactly two costumes, and each costume is liked by exactly two people. Prove that it is possible to match each person with a costume they like. 1
欢迎咨询51作业君