18.314     PROBLEM SET 2    (due Tuesday October 04, 2005)

Problems 1-5: [Bona] p.78 #32, p.102 (#18, #20, #22), p.123 #30

Problem 6: For positive integers n, k, show that the sum of 
  binomial coefficients C(n,0) + C(n+1,1) + C(n+2,2) + ... + C(n+k,k) 
  (diagonal entries in the Pascal triangle) is a certain binomial 
  coefficient.

  Here and in Problem 12, C(a,b) is the binomial coefficent "a choose b".

Problem 7:  Prove that the number of compositions of n with all 
  odd parts equals the Fibonacci number.
 
Problem 8:  Show that the numbers F(n,r) of compositions 
  on n with all parts >= r satisfy the "generalized 
  Fibonacci relation": F(n,r) = F(n-1,r) + F(n-r,r), for n>r.
  Deduce that the numbers F(n,2) are the Fibonacci numbers.

Problem 9: Find a bijection between partitions of n with
  odd parts and partitions of n with distinct parts.

Problem 10: A self-conjugate partition is a partition lambda such 
  that lambda' = lambda.  For a nonnegative integer r, calculate 
  the number of self-conjugate partitions with all parts <= r.  

Problem 11: (Bonus) The amazon (also known as the maharaja and as the 
  cavalry queen) is the chess piece that can move as the queen and as 
  the knight.  Find the maximal number of nonattacking amazons that can 
  be placed on the 6 x 6 chessboard?  What can you say about an n x n 
  chessboard?

Problem 12: (Bonus) Describe all integers a, b, c, d such that 
  the inequality C(a,b) C(a,c) >= C(a,b+d) C(a,c-d) holds.
  What about a more general inequality of the form 
  C(a,b) C(c,d) >= C(a+e, b+f) C(c-e,d-f) ?