18.314     PROBLEM SET 3    (due Tuesday October 25, 2005)


Problem 1 ([Bona, p. 139, #15]).  Prove combinatorially (that is, not using 
   the formula for D_n) that D_{n+1} = n(D_n + D_{n-1}).
   Here D_n is the number of derangements in S_n.

Problem 2 ([Bona, p. 139, #19]).  How many positive integers are there that
   are not larger than 1000 and are neither perfect squares nor perfect cubes?

Problem 3.  Let p_1,...,p_k be different prime numbers.  Find a simple 
   expression for the number of integers i such that 1 <= i <= p_1 p_2 ... p_k 
   and i is not divisible by p_j, for j = 1,...,k.

Problem 4.  Let Q_n be the number of permutations w in S_n such 
   w_{i+1} is not equal to w_i + 1 for any i.  The number Q_n is related 
   to the derangement number D_n as Q_n = D_n + D_{n-1}.
   (A) Prove this identity using the formulas for D_n and Q_n.
   (B) Find a bijective proof.

Problem 5. Find an explicit expression for a_n if 
   a_{n+2} = 5 a_{n+1} - 6 a_n, for n >= 0, and a_0 = 0, a_1 = 1.

Problem 6. Find an explicit expression for a_n if 
   a_{n+2} = 5 a_{n+1} - 6 a_n + 2n - 1, for n >= 0, and a_0 = 0, a_1 = 1.

Problem 7. ([Bona, p. 170, #26]).  Find an explicit formula for a_n if
   a_{n+1} = (n+1) a_n + 2 (n+1)! if n >= 0, and a_0 = 0. 

Problem 8.   Let f_n be the number of ways to subdivide n children into groups
   of 2, 3, or 4 children.  (Also set f_0 = 1.)  Find a simple expression for the 
   exponential generating function f_0 + f_1 x/1! + f_2 x^2/2! + + f_3 x^3/3! + ...

Problem 9. ([Bona, p. 170, # 28]).  Find the exponential generating function
   D(x) = D_1 x + D_2 x^2/2! + D_3 x^3/3! + ... for the numbers of derangements D_n.

Problem 10.  Find the number of ways to place 8 nonattacking rooks on the 8 x 8 
   chessboard with forbidden positions marked by X's below:

      _______________________________
     | X | X | X |   |   |   |   |   |
     |___|___|___|___|___|___|___|___|
     |   |   |   | X |   |   |   |   |
     |___|___|___|___|___|___|___|___|
     |   |   |   | X |   |   |   |   |
     |___|___|___|___|___|___|___|___|
     |   |   |   | X |   |   |   |   |
     |___|___|___|___|___|___|___|___|
     |   |   |   |   | X | X | X |   |
     |___|___|___|___|___|___|___|___|
     |   |   |   |   |   |   |   | X |
     |___|___|___|___|___|___|___|___|
     |   |   |   |   |   |   |   | X |
     |___|___|___|___|___|___|___|___|
     |   |   |   |   |   |   |   | X |
     |___|___|___|___|___|___|___|___|


Problem 11 (bonus).   Consider the infinite continued fraction 

   c(x) =      1
           -------------------------
            1 -   x
                 -------------------
                  1 -   x
                      --------------
                       1 -  x                    
                           ---------
                            1  - ...
                      
   (A) Evaluate this continued fraction at x = -1.

   (B) Expand this continued fraction as the power series 
       c(x) = c_0 + c_1 x + c_2 x^2 + ...
       Find a simple expression for c_n.

Problem 12 (bonus).  Let a_n be the sequence defined by 
   a_{n+2} = (a_{n+1}^2 + 1)/a_n, for n >= 0, and a_0 = a_1 = 1.
   Find a simple expression for a_n.


Problem 13 (bonus).  (A) Find a bijection between Catalan paths of
   length 2n and parenthesations of n+1 letters (or binary trees with
   n+1 leaves).

   (B)  A permutation w in S_n is called 123-avoiding if there are no
   i < j < k such that w_i < w_j < w_k.  A permutation w in S_n is 
   called 132-avoiding if there are no i < j < k such that w_i < w_k < w_j.
   Show bijectively that the number of 123-avoiding permutations in S_n
   equals the number 132-avoiding permutations equals the Catalan number C_n.