Math 491, Problem Set #20: Solutions 1. (from unpublished work of Douglas Zare) Let Gm,n be the directed graph with vertex set cfw_(i, j) Z Z : 0 i m, 0 j n, with an arc from (i, j) to (i , j ) iff (j - j, i - i) is (1, 0), (0, 1), or (1, 1). (a) For any
Math 491, Problem Set #19: Solutions
1. Let P (n) and Q(n) denote the numerator and denominator obtained when the continued fraction x1 + (y1 /(x2 + (y2 /(x3 + (y3 / + (yn-2 /(xn-1 + (yn-1 /xn ) ) is expressed as an ordinary fraction. Thus P (n) and Q(n)
Math 491, Problem Set #17: Solutions
1. Let p(n) be the number of unconstrained partitions of n if n 0, and 0 othewise, so that p(n) = p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-12)+p(n-15)-+ . . . for all n > 0. Use the recurrence for p(n) to compute the last digit
Math 491, Problem Set #15 Solutions
(a) Let An be the average number of times that a 2n-step Dyck path returns to the origin (counting (2n, 0) as a return but not (0, 0), so that A0 = 0, A1 = 1, A2 = 3/2, and A3 = 9/5. Use Maple to compute An for various
Math 491, Problem Set #14: Solutions
1. Consider the subset of the square grid bounded by the vertices (0, 0), (m, 0), (0, n), and (m, n), and let q be a formal indeterminate. Let the weight of the horizontal grid-edge joining (i, j) and (i + 1, j) be q j
Math 491, Problem Set #13: Solutions
1. An "augmented Aztec diamond of order n" is a subset of the square grid formed by symmetrically stacking rectangles of height 1 and respective widths 2, 4, 6, . . . , 2n - 2, 2n, 2n, 2n, 2n - 2, . . . , 6, 4, 2 (note
Math 491, Problem Set #11 (Solutions)
1. Let cn be the number of domino tilings of a 3-by-2n cylinder, obtained by gluing together the left and right sides (of length 3) of a 3-by-2n rectangle. Express the generating function c1 x + c2 x2 + c3 x3 + . . .
Math 491, Problem Set #12: Solutions
1. Define the diagonal of a two-variable generating function F (x, y) =
m,n
am,n xm y n
as the generating function D(t) =
n
an,n tn .
It is a theorem (which we will not have time to prove) that the diagonal of any two-
Math 491, Problem Set #7: Solutions
1. One basis for the space of polynomials of degree less than d is the monomial basis 1, t, t2 , ., td-1 . Another is the shifted monomial basis 1, (t + 1), (t + 1)2 , ., (t + 1)d-1 . Call these bases u1 , ., ud and v1
Math 491, Problem Set #6: Solutions
1. There is a unique polynomial of degree d such that f (k) = 2k for k = 0, 1, ., d. What is f (d + 1)? What is f (-1)? Suppose g(k) is a polynomial of degree m 1, so that its sequence of mth differences is constant. If
Math 491, Take-home Midterm Do all of the following problems: 1(a) (20 points) For n 1, let an be the number of ways to put pennies on the cells of a 2-by-n rectangle (at most one penny per cell) so that no two pennies are horizontally or vertically adjac
Math 491, Problem Set #3: Solutions
1. (a) Consider the sequence 1, 1, 1, 3, 3, 7, 9, 17, 25, . satisfying the initial conditions a0 = a1 = a2 = 1 and the recurrence relation an = 2an-2 + an-3 . Write the generating functions A(x) = an xn n=0 as a rationa
Math 491, Problem Set #1: Solutions
1. (a) Write and run a program to compute f (n) = Heres one thatll do the job:
2 n k n k=0 (1) k .
f := proc(n) local k; RETURN(sum(-1)^k*binomial(n,k)^2,k=0.n); end; Notice that the variable k is declared to be local;
Math 491, Take-home Final (due December 18, 11:00 a.m.) Read the instructions at the end of this exam, and then do all of the following problems: 1. A Motzkin path is a nite path that visits the successive points (0, m0 ), (1, m1 ), (2, m2 ), . . ., and (
Math 491, Problem Set #11 (due 10/21/03)
1. Let cn be the number of domino tilings of a 3-by-2n cylinder, obtained by gluing together the left and right sides (of length 3) of a 3-by-2n rectangle. Express the generating function c1 x + c2 x2 + c3 x3 + . .
Collect homework, and comments on how to improve the homework Any questions about the homework or the material? TODAY: Paths in DAGs Counting paths by multiplying matrices Domino tilings again Weighted enumeration A finite DAG (directed acyclic graph) is
TODAY: Short recurrence for counting lattice paths by q-weight Number partitions Short recurrence for counting lattice paths by q-weight Warm-up: combinatorial interpretation of (a choose b) b = a (a-1 choose b-1) Consequence: (a choose b) = (a/b) (a-1 ch
Any questions about the exam? Homework? Maple? Final exam vs. final project Return exam Give overall comments: 1. Read everything before you start. Some of you didn't realize that when I ask for a formula for a sequence, I want a closed form formula, not