Chapter 1, Problem 1.15TE

Let $H_n\left(n\right)$ be the number of vectors $x_1,\mathrm{...},x_k$ for which each $x_i$ is a positive integer satisfying $1\le x_i\le n$ and $x_1\le x_2\le \mathrm{...}\le x_k$.

a. Without any computations, argue that $\begin{array}{l}{H}_1\left(n\right)=n\\ H_1\left(n\right)={\displaystyle \underset{j=1}{\overset{n}{\sum }}{H}_{k-1}\left(j\right)k>1}\end{array}$

Hint: How many vectors are there in which $x_k=j$?

b. Use the preceding recursion to compute $H_3\left(5\right)$.

Hint: First compute $H_2\left(n\right)$ for $n,=1,2,3,4,5$.