Recommended questions: a-d 4f 5 9a a 27.

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1 Sheet Recommended questions: 2 3 4a-d 4f 5 9a a 27 Recommended reading for this assignment: Aigner Chapter, Chapter 2-24 Formal power series Question Let R be a commutative ring containing the rationals and consider R[[X, Y ]] Is Y X a formal power series in R[[X, Y ]]? What about exp(exp(x? Explain Question 2 Prove the following statements where R is a commutative ring with characteristic zero (a (b (c (d If R is an integral domain then R[[ X]] is also an integral domain If A, B R[[ X]] satisfy j A( X = j B( X and A( 0 = B( 0, and R has no divisors of 0, then A = B If A R[[ X]] where A( 0 = a n and a R is invertible, then there is a unique B R[[ X]] with B n = A and B( 0 = a given by B( X = a ( /n (a n A( X Suppose A R[[X]] has A(0 = 0 Prove that A has a compositional inverse if and only if A(0 0 Question 3 Let R be a commutative ring with unity R and characteristic zero containing the rational numbers Derive each of the following from the definitions of formal power series: Show that the binomial theorem for positive integer exponents implies exp(x + Y = exp(x exp(y Show conversely that the binomial theorem for positive integer exponents follows from exp(zx + ZY = exp(zx exp(zy 2 Binomial sums Question 4 Prove each of the following identities for m, n N, or determine the value of the sum:

2 ( (a n 0 n/2 (b n ( ( n 3 = 2 3 n/2 cos(nπ/6 (c ( n m ( i=0 i( i = n+m (d n ( n cos(θ = (2 cos(θ/2 n cos(nθ/2, θ R (e n ( n+ 2 = 2 n (f n ( n+ 2 n (g ( n+ 2 ( m+2( = ( n + m (h n ( n = n+ ( n+ 2 n Reciprocals, integrals, derivatives Question 5 Prove the following identities in C[[X]]: (a (b + 2X + 3X 2 + 2X 3 + X = 4 ( 2X + X 2 = n=0 m=0 j= j ( j ( j i X i+j 2 i i= n ( ( m + n m + n m ( X n (c Deduce from (b that for, n N, ( 2 + n = 2 n ( ( m + n m + n m m=0 Question 6 Let n N {0} Prove that if [X n ]A(X = a n then [X n ] A(X n X = a Prove that if H n is the nth Harmonic number then n= H n X n log( X = X Determine also a closed form formula for a formal power series whose coefficient of X n is ( n H n 2

3 Question 7 Determine a n, = [X n Y ]A(X, Y for n, N {0}, where A(X, Y = XY Y Can you thin of an identity or recurrence satisfied by a n,? Question 8 Show that X 2 + X = 3 + X2 X and then show for n N that ( 3n ( = 3n 2 Question 9 (a Find A C[[X]] satisfying A(0 = and X 2 A(X + (X A(X + = 0 For z C, what is A(z? (b Prove that for n N, Xn ( X n+ n! n X = ( X n+ + ( 2n n X n n ( 2n X +n + n ( X + 4 Recurrence equations Question 0 For n N, solve for a n C: (a (n + a n+ = a n + a n where a 0 = a = (b a n+ = n n a a n with a 0 = 0 and a = Question For n N, prove that at most n log 2 n + n rounds are needed to sort a set of n distinct elements with the following algorithm: Divide set into singleton subsets which are each considered to be sorted 2 Repeatedly merge sorted subsets to produce new sorted subsets until there is only one sorted subset remaining 3

4 For instance the (unsorted set {4, 2, 7, 9, 3, 6, 0, 5} breas into 4, 2, 7, 9, 3, 6, 0, 5 Round of the algorithm would produce 24, 79, 36, 05 Round 2 produces 2479 and 0356 and the final round produces , which is sorted [Hint: show that the number of rounds is at most a n, where a n = n + a n/2 + a n/2 for n 2 and a = 0] Question 2 Let a n be the number of tilings of a 3 n grid with dominoes Prove that for n 4, a n = 4a n 2 a n 4 Determine a formal power series A(X in closed form with [X n ]A(X = a n for n 5 Compositions Question 3 Let, m N with < m, and let n = ( mj Prove that the number of solutions (x, x 2,, x to x + x x = n such that x i i mod m for i [] is ( +j j Question 4 Let N N and let p n p n 2 2 p n be the prime factorization of N where n i N for i [] Find the number of ordered factorizations of N into positive integers Question 5 Let n, N Show that the number of subsets of [2n] of size such that no pair of consecutive elements (in the increasing order of the elements differ by exactly two is ( ( n i + n + i + i i i=0 Question 6 Determine an explicit formula for the formal power series in C[[X, Y ]] whose coefficient a n, of X n Y is the sum over all compositions of n with parts of the product of the parts of the composition For instance, a 4,2 = = 0 Question 7 Let, n N and let S = { n = (n, n 2,, n : 0 n < n 2 < < n < n} Show that n S X i= ( n i i = X (n X 4

5 Deduce that given N, each positive integer N has a unique representation N = with = (n, n 2,, n S n i= ( ni i 6 Strings Question 8 Find the generating function for binary strings by length in which (a the substring 00 never occurs and (b an odd bloc of 0s is never followed by an odd bloc of s Determine a recurrence equation for the number of strings of length n of each type in (a and (b Question 9 Let n N and let S be the set of binary strings such that each bloc of 0s is followed by a bloc of s of the same parity Show that the number of strings of length n in S is 2 3 n/2 for n 2 Question 20 Let m, n N (a What is the average number of blocs of s in a binary string of length n? (b Prove that there are ( n+ n 2m strings of length n such that 0 appears m times Question 2 For, n N, let a n, be the number of binary strings of length n in which 00 occurs times as a substring Prove that a n, = [X n Y ] 2X (Y X 4 Question 22 Prove that the equation {0,, 2} = {0} (({, 2} \ε{0}{0} {, 2} uniquely creates all ternary strings For n N, determine the number of strings of length n with no substring of two consecutive 2s Question 23 Let, n N Prove that the number of -ary trees with n vertices is ( n + n + n 5

6 Question 24 Determine (a an implicit equation for the exponential generating function of rooted Cayley trees which have no vertex of degree exactly two and n vertices and (b the number of binary rooted Cayley trees ie rooted Cayley trees in which no vertex has degree more than three and the root vertex has degree at most two Question 25 (a A pyramid consists of rows of contiguous unit squares one on top of the other, so that each successive row has no unit square above the squares at the ends of the preceding row Let, n N How many pyramids have width n and height? (b A polyomino consists of a union of columns of unit squares in Z 2 such that any two adjacent columns have at least one unit square side length in common How many polyominoes can be made out of five squares? 7 Lagrange inversion Question 26 Show that exp(xy X = X (Y +! exp( X Question 27 Let n N and let f(z, w : C 2 C be defined by f(z, w = ( z n+ z w Determine using Lagrange s inversion formula n f z n at z = Question 28 Let m, n, p N Prove ( ( ( m + n n + p p + m ( = m + n + p + (m + n + p! m!n!p! 6