% Christopher Hanusa
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\begin{center}\large 
MATH 636, Fall 2014

{\sc Homework 7}

due 5:00{\sc pm} on Tuesday, December 2.
\end{center}

\smallskip\noindent
{\bf Follow the posted homework guidelines} when completing this assignment.  \\ Please {\bf \color{red}only} consult with your classmates or professor to discuss the problem set.

\medskip\noindent
These four questions are each worth five points each.  Continue work on your project.

\begin{enumerate}


\item[\bf 7-1.] 
Recall that a {\bf Dyck path of length $n$} is a lattice path from $(0,0)$ to $(n,n)$ that stays above the line $y=x$.)
\begin{enumerate}
\item Find and list the 14 Dyck paths of length 4 and the 14 multiplication schemes for 5 variables. 
\item Use the Catalan bijections from class to determine which Dyck path corresponds to which multiplication scheme.\end{enumerate}

\item[\bf 7-2.] Use a bijection to show that sequences $1\leq a_1\leq a_2\leq \cdots \leq a_n$ of length $n$, where each $a_i\leq i$ are also counted by the Catalan number $C_n$. For example, when $n=3$, the five sequences are $111$, $112$, $113$, $122$, and $123$. 

[{\em Hint: Look at the boxes to the left of a Dyck path.}]

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\item[\bf 7-3.] Two combinatorial interpretations of the $q$-binomial coefficients are given on page 128 of the course notes. 
\begin{enumerate}
\item Show that for the permutations $\pi$ of the multiset $\{1^2,2^3\}$, $\sum_{\pi\in S_{2,3}} q^{\mathsf{inv}(\pi)}=\qbinom{5}{3}$.
\item Show that for the set of lattice paths $P$ from $(0,0)$ to $(2,3)$, $\sum_{P\in \mathcal{P}} q^{\mathsf{area}(P)}=\qbinom{5}{3}$.
\end{enumerate}

\item[\bf 7-4.] Let $\mathcal{C}_n$ denote the set of compositions of $n$. \newline 
For any composition $c$, define the statistic $\mathsf{parts}(c)$ to be the number of parts of $c$.  \newline
[{\em In other words, if $c$ is the composition $c_1+c_2+\cdots+c_k$, then $\mathsf{parts}(c)=k$.}] \vspace{-.08in}
\begin{enumerate}
\item Compute $\displaystyle f_n(q)=\sum_{c\in\mathcal{C}_n} q^{\mathsf{parts}(c)}$.\vspace{-.1in}
\item Use your answer to part (a) to show directly $\displaystyle\lim_{q\to 1} f_n(q)=2^{n-1}$.  \vspace{-.10in}
\end{enumerate}
[{\em Note: We expect part (b) to be true because we know there are $2^{n-1}$ compositions of~$n$, and part~(a) is constructing a $q$-analog.}]
\end{enumerate}


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