Due: Nov. 29, 2004.
1a. Modify the procedure Wilf(n,SetP) in VATTER to allow any set of patterns of any length (not just patterns of length 3).
1b. Find five different integer sequences that count Wilf classes for various sets of forbidden patters, and look for them in Sloane.
1c. (Optional, 5 dollar prize) Conjecture necessary and sufficient conditions for a permutation of {1, ..., n} to be both 123- and 132- avoiding, then prove them mathematically (by induction or otherwise), and deduce from them the theorem that the cardinality is 2**(n-1).
2a. Modify the program Ortho(n,x,w,A,B), to write a program OrthoU(n0,n,x,T) in the Maple package ORTHO so the input, instead of being a weight-function w(x), is an expression,T, of (the discrete variable n), that defines an "umbra" x**n GOES TO T(n) and outputs the polynomial of degree n0 such that when multiplied by x^i (i less than n0) and applying the umbra, gives zero. In particular OrthoU(5,n,x,1/(n+1)); should give the same output at Ortho(5,x,1,0,1);
2b. By using PolF or Rat or whatever, conjecture an explicit expression for OrthoU(n0,n,x,n!);
2c. (optional, open-ended): experiments with various T's and see if you can get explicit expressions for the corresponding orthogonal polynomials.
3. Using the package ORTHO , write a program that inputs an integer n and outpus the n points in (0,1) x[1], ..., x[n] and the n numbers A[1], ..., A[n] such that int(f,x=0..1)=A[1]f(x[1])+A[2]f(x[2])+ ... + A[n]f(x[n]) for every polynomial f(x) of degree <=2n-1. Then write another program that implements this quadrature, and compare the appx. for various functions, e.g. sin(x), cos(x), exp(x), and higher powers.
4. Do the analog of GOLAY for the other Platonic solids (except for the icosahedron), and see if you get any good error-correcting codes.
5. Write a program that inputs integers a and b, and a positive integer L, and outputs the maximum length of consecutive digits in the first L digits of the decimal expansion of exp(a)*Pi^(b). (Hint: set DIGITS:=L+5.) In the range a,b between 1 and 100, who is the winner?, and what is the record?
6. Write a Maple procedure that inputs an integer L and simulates L throwings of a fair-(cubical)-die, and outputs the sum of the numbers. Then write another program that does it K times, and find the average and variance. Compare the oupputs (or latge K, say K=10000) to the theoretical average and variance.