PROBLEMS

(1) Write a MAPLE program to compute p(n) using the Euler
    recurrence. Use it to compute p(101).
    There is a table of p(n) on pp.238-240 of the text
    so you can check your program.

(2) Use MAPLE to find the generating function for
    p   (n) which is the number of partitions of n into parts
     4,4    less than or equal to 4 in which each part occurs
            at most 4 times.
    Your answer should be a polynomial in q.
    Hence find
    p   (20).
     4,4

(3) (i) Compute the GF of p(D,n) at least up to q^20.
        Hence find p(D,14).
        [HINT: Use the infinite product form of the GF]
    (ii) Verify p(D,n)=p(O,n) for n=14
        by computing the partitions of 14 into distinct parts and
        an the partitions of 14 into odd parts.    
    [HINT: See Example 3 of MAPLE EXAMPLES.]

(4) [MACMAHON]
    Let M[1](n) denote the number of partitions of n into parts,
    each larger than 1, such that consecutive integers do not
    appear as parts. Let M[2](n) denote the number of partitions on n
    in which no part appears exactly once. Prove that

    M[1](n) = M[2](n)  for all n.

    HINT: There is a simple bijection between the two sets
           of partitions.

(5) [MACMAHON]
    Let M[3](n) denote the number of partitions of n into parts
    into parts not congruent to 1 or 5 mod 6.  Prove that

    M[2](n) = M[3](n)  for all n.

    HINTS:
     (i) First write the GF of M[3](n) as an infinite product.

     (ii) Show that 1 + x^2 + x^3 + ...  = (1 - x^6)/((1 - x^2)(1 - x^3))
          for |x|<1 and hence write the GF of M[2] as an infinite
          product.

(6) [EULER]
    Prove that the absolute value of the number of (unrestricted)
    partitions of n with an odd number of parts over those with
    and even number of parts equals the number of parts into distinct
    odd parts.

    HINT: If  
    prod( (1-q^(2*n-1)), n=1 .. infinity) = sum( a[n]*q^n, n=0 .. infinity)
    then show that 
    prod( (1+q^(2*n-1)), n=1 .. infinity) = sum( |a[n]|*q^n, n=0 .. infinity)

The url of this page is http://qseries.org/fgarvan/qs/summer2005/hw/probs.html.
Created by F.G. Garvan (fgarvan@ufl.edu) on Tuesday, May 10, 2005.
Last update made Tue May 10 10:09:37 EDT 2005.

fgarvan@ufl.edu