18.314     PROBLEM SET 4    (due Tuesday November 01, 2005)


Problem 1.  Let s_n be the number of lattice paths from (0,0)
   to (n,n) with the steps R=(1,0), U=(0,1), and D=(1,1) that never
   go below the line y = x.  For example,  a_2 = 6, corresponding 
   to the paths (U,R,U,R), (U,U,R,R), (U,R,D), (U,D,R), (D,U,R), (D,D).
   Calculate the generating function 1 + s_1 x + s_2 x^2 + ... 
   for these numbers.

Problem 2.  A man stands a few steps away from the edge of a cliff.  With 
   the probablity p he makes a right step away from the edge and with the 
   probability (1-p) he makes a left step towards the edge.  He continues 
   to walk in this random fashion.  Say, the probability of two right steps 
   followed by a left step is pp(1-p).  If the man steps over the edge of 
   the cliff, he falls down and dies.

   (A) Assume that originally the man stands right on the edge of the cliff.
   Show that the probability P(n,death) that the man dies after making 
   exactly 2n+1 steps is C_n p^n (1-p)^(n+1), where C_n is the Catalan number.

   (B) Same assumption as in (A).  Calculate the probability that the man 
   survives after making indefinitely many steps.  
   (Hint:  This probablity is equal to 1 - P(0,death) - P(1,death) - ...)
   Find the condition on p that ensures that the man has a nonzero chance 
   of survival.

   (C) Assume that originally the man stands k steps away from the edge
   of the cliff.  Calculate the probability P_k of his survival in this 
   case.  For example,  P_0 is the probability calculated in (B) and
   P_{-1} = 0 (the man is already over the edge).
   
   (D) Find a recurrent relation for the sequence P_k from (C).

Problem 3.  It was shown in class that the Motzkin numbers M_n are
   related to the Catalan numbers C_n as M_n = sum C(n,2k) C_k over 
   k = 0,...,[n/2], where C(n,m) is the binomial coefficient n choose m.  
   Show that  C_n = sum (-1)^k C(2n,k) M_{2n-k} over k = 0,...,2n.

Problem 4.  A triangulation of a polygon is a way to subdivide it 
   into triangles by noncrossing diagonals.  The number of all 
   triangulations of the (n+2)-gon equals the Catalan number C_n.
   Find the number of centrally symmetric triangulations of
   the regular 2n-gon.