18.314 Fall 2007 Problem Set 4. Due Thursday, November 01. 1. Let a_n be the sequence given by a_n = 5 a_{n-1} - 6 a_{n-2}, for n>=2, a_0 = 0, a_1 = 1. (a) Find the ordinary generating function for the sequence a_n. (b) Find the exponential generating function for the sequence a_n. (c) Give an explicit expression for a_n. 2. Let b_n be the sequence given by the recurrence relation b_n = 5 b_{n-1} - 6 b_{n-2} + 4n, for n >=2, and b_0 = 6 , b_1 = 12. (a) Find the exponential generating function for the sequence b_n. over n >= 0). (b) Find an explicit expression for b_n. 3. Find two constants A and B such that the sequence c_n = 4^n + 3^n satisfies the recurrence relation c_n = A c_{n-1} + B c_{n-2}. 4. For a partition lambda = (lambda_1, lambda_2,...) let D(lambda) be the size of the maximal square that fits inside the Young diagram of lambda. (This square is called the Durfee square.) For example, for lambda = (7,7,5,3,3,1), D(lambda) = 3. The following picture shows the corresponding Young diagram, where the Durfee square is marked by o's. o o o . . . . o o o . . . . o o o . . . . . . . . . Let p(n,r) denote the number of partitions lambda of n with D(lambda)= r. Find the ordinary generating function ___ \ /__ p(n,3) x^n. n>=0 5. Let f_n be the number of graphs G on n nodes labelled by integers 1,...,n such that each connected component of G is a path, that is a sequence of vertices connected by edges i--j--k--...--l. (We allow connected components with a single vertex.) Also assume that f_0 = 1. For example, for n = 3, there are 7 such graphs (all graphs with 3 labeled vertices, except K_3). Find the exponential generating function ___ \ /__ f_n x^n/n! n>=0 6. For two vertices a and b in a graph, let the distance d(a,b) be the number of edges in a shortest path between a and b. Let g_n be the number of trees on n vertices labelled 1,2,...,n such that, for any vertex i, we have d(1,i) <= 2. Assume that g_0 = 1. Find the exponential generating function ___ \ /__ g_{n+1} x^n/n! n>=0 7. Construct a bijection between Dyck paths with 2n edges and triangulations of the (n+2)-gon. 8. Show that the number of Dyck paths of length 2n that start with exacty k up steps equals the number of Dyck paths of length 2n that have exactly k+1 points on the horizontal x-axis. A bijective proof is preferable. For example, for n = 3 we have /\ /\ /\ /\/\ / \ Dyck path /\/\/\ /\/ \ / \/\ / \ / \ # initial up steps 1 1 2 2 3 # points 4 3 3 2 2 on x-axis 9*. (Bonus problem) Show that the number integer vectors (x_1,...,x_n) such that x_1 + x_2 + ... + x_n = 0 mod (n+1) 0 <= x_i <= n, for i = 1,...,n x_1 <= x_2 <= ... <= x_n equals the Catalan number C_n. For example, for n = 3, we have the following C(3) = 5 vectors: (0,0,0), (0,1,3), (0,2,2), (1,1,2), (2,3,3). 10*. (Bonus problem) Let T be a tree on n nodes labeled 1,...,n. Let us direct each edge (i,j) of T from i to j if i