Chapter 1, Problem 1.16TE

Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That Is. the outcome partitions the players into groups. with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let N(n) denote the number of different possible outcomes. For instance, $N\left(2\right)=3$, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first

a. List all the possible outcomes when $n=3$.

b. With N(0) defined to equal 1, argue, without any computations, that $N\left(n\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}N\left(n-i\right)}$

Hint: How many outcomes are there in which i players tie for last place?

c. Show that the formula of part (b) is equivalent to the following: $N\left(n\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(\begin{array}{c}n\\ i\end{array}\right)N\left(i\right)}$

d. Use the recursion to find N(3) and N(4).