Snevily, Hunter S.

Roller Coaster permutations ★★★

Author(s): Ahmed; Snevily

Let $S_n$ denote the set of all permutations of $[n]=\set{1,2,\ldots,n}$. Let $i(\pi)$ and $d(\pi)$ denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in $\pi$. Let $X(\pi)$ denote the set of subsequences of $\pi$ with length at least three. Let $t(\pi)$ denote $\sum_{\tau\in X(\pi)}(i(\tau)+d(\tau))$.

A permutation $\pi\in S_n$ is called a \emph{Roller Coaster permutation} if $t(\pi)=\max_{\tau\in S_n}t(\tau)$. Let $RC(n)$ be the set of all Roller Coaster permutations in $S_n$.

\begin{conjecture} For $n\geq 3$, \begin{itemize} \item If $n=2k$, then $|RC(n)|=4$. \item If $n=2k+1$, then $|RC(n)|=2^j$ with $j\leq k+1$. \end{itemize} \end{conjecture}

\begin{conjecture}[Odd Sum conjecture] Given $\pi\in RC(n)$, \begin{itemize} \item If $n=2k+1$, then $\pi_j+\pi_{n-j+1}$ is odd for $1\leq j\leq k$. \item If $n=2k$, then $\pi_j + \pi_{n-j+1} = 2k+1$ for all $1\leq j\leq k$. \end{itemize} \end{conjecture}


Snevily's conjecture ★★★

Author(s): Snevily

\begin{conjecture} Let $G$ be an abelian group of odd order and let $A,B \subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered $A = \{a_1,\ldots,a_k\}$ and $B = \{b_1,\ldots,b_k\}$ so that the sums $a_1+b_1, a_2+b_2 \ldots, a_k + b_k$ are pairwise distinct. \end{conjecture}

Keywords: addition table; latin square; transversal

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