18.314  Fall 2007

Problem Set 2.  Due Thursday, October 4.


1.  For n >= k >= 0, show that 

   C(k,k) + C(k+1,k) + C(k+2,k) + ... + C(n,k) = C(n+1,k+1).

(Here C(n,k) is "n choose k".)


2.  Find a closed expression for the sum of the entries of 
the Pascal triangle inside the upper n x n rhombus.  
For example, for n = 3, you need to sum the entries:

                     1
                   1   1
                 1   2   1
               .   3   3   .
             .   .   6   .   . 


3.  Find a closed expression for the number of lattice paths P 
on the plane with the steps (1,0) "right" and (0,1) "up" such that 
P starts at the point (0,0), ends at the point (2n,n), 
and P always stays in the area {x >= y} (below the line x=y).


4.  Find the number of compositions of 25 into 5 odd parts.


5.  Find a bijection between the following 2 sets:
   A = {compositions of n with all parts equal to 1 or 2}
   B = {compositions of n+2 with all parts greater than 1}.


6(*). (optional)  Find a bijection between the following 2 sets:
   A = {compositions of n with all parts equal to 1 or 2}
   C = {compositions of n+1 with all odd parts}


7.  Let p(n,k) be the number of partitions of n with all parts 
greater then or equal to k.  For n, k >= 1, show that 

      p(n+k,k) = p(n,k) + p(n+k, k+1).

(optional part)  Use this formula, a computer, and your favorite 
tool (maple, mathematica, C, C++, ...) to calculate p(1000,1).


8.  Find a closed expression for the number of set partions of
the set {1,2,...,3n} into 3 blocks with n elements.
For example, for n=1, there is only one such set partition 1|2|3.
(We don't care about ordering of the blocks.  Say, 2|1|3 is the 
same thing as 1|2|3.)


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


10.  For n and k, let R(n,k) be the number of set partions
of [n] with k nonempty blocks such that there is no block
that contains two adjacent entries i and i+1.  For example, 
R(5,3) = #{13|24|5, 13|25|4, 14|25|3, 14|35|2, 24|35|1,  
15|24|3, 135|2|3} = 7.  Express the number R(n,k) in terms 
of the Stirling numbers of the second kind.


11.  Prove the following identity for the signless Stirling 
numbers of the first kind:

    1 c(n,1) + 2 c(n,2) + 3 c(n,3) + ... + n c(n,n) =
                    = n! (1 + 1/2 + 1/3 + 1/4 + ... + 1/n)