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\begin{center}
{\Large NMU Math \& CS Department} \\
\vspace{0.5cm}
{\Large Problem of the Month, November 2024}
\end{center}

\vspace{1mm}

\paragraph{} Consider the strange sequence of functions $F_{1},F_{2},F_{3},F_{4}...$ given by:

\begin{align*}
&F_{1}(a)=a \\
&F_{2}(a,b)=a+b \\
&F_{3}(a,b,c)=(a+b,a+c,b+c) \\
&F_{4}(a,b,c,d)=(a+b,a+c,a+d,b+c,b+d,c+d) \\
& \hspace{1cm} \vdots \\
&F_{n}(z_{1},z_{2},...,z_{n})=(z_{1}+z_{2},z_{1}+z_{3},...,z_{1}+z_{n},z_{2}+z_{3},...,z_{n-1}+z_{n}).
\end{align*}

\paragraph{} The first thing you may notice is that for $n \geq 2$, if there are $n$ coordinates on the left side, there will be $(n-1)+(n-2)+...+1=\frac{n(n-1)}{2}$ coordinates on the right side. Therefore you may interpret these functions as mappings between tuples of integers, and we will in fact consider them as mappings between \textbf{tuples of positive integers}.

$$F_{n}:\mathbb{N}^{n} \longrightarrow \mathbb{N}^{n(n-1)/2}, \text{ for }n \geq 2.$$

\paragraph{} For example we have $F_{3}:\mathbb{N}^{3} \rightarrow \mathbb{N}^{3}$, and $F_{4}:\mathbb{N}^{4} \rightarrow \mathbb{N}^{6}$, and $F_{4}(1,2,3,4)=(3,4,5,5,6,7)$.

\paragraph{} \textbf{The Warm up.} Show that if $X,Y \in \mathbb{N}^{n}$ are two $n$-tuples with ``the same elements'' up to reordering, then $F_{n}(X)$ and $F_{n}(Y)$ have ``the same elements'' up to reordering. As an example when $n=3$: $(1,2,3)$ and $(2,3,1)$ have the same elements up to reordering, and we see that $F_{3}(1,2,3)=(3,4,5)$ and $F_{3}(2,3,1)=(5,3,4)$ have the same elements up to reordering.

\paragraph{} \textbf{The Problem.} Show that the following statement is true precisely when $n \neq 2^{k}$.

\begin{equation*}
( F_{n}(X),F_{n}(Y) \text{ same elements up to reordering}) \Rightarrow (X,Y \text{ same elements up to reordering})
\end{equation*}

\paragraph{} \textit{Hint}: Given any tuple $W=(a_{1},...,a_{k})$, try to make use of the associated polynomial $p_{W}(x)=x^{a_{1}}+\cdots+x^{a_{k}}$. For example $p_{(1,2,3)}(x)=x+x^2+x^3$.

\newpage

\paragraph{} The NMU Mathematics and Computer Science Department invites you to participate in the 2024/2025 Problem of the Month contest to have some fun and get a little recognition. There are paper copies of the problems available at the department front desk if you'd like.

\paragraph{} \textbf{Rules:} Anyone is welcome to submit a solution, and all correct solutions will be recognized; but only undergraduates enrolled in coursework at NMU are eligible to win the prize at the end of the year. The first student to submit a correct solution is the winner of the month. The top problem solver for the academic year will receive a fabulous prize, and be recognized at the department year-end celebration. You must write clear and complete solutions to the problems. You must include your name, NMU IN, address, phone, email, and exact date/time of your submission. You may either submit in person at the department front desk (use a staple if there are multiple pages), or email your solution in \texttt{pdf} format to \texttt{darowe@nmu.edu}.



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