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\begin{document}

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\begin{center}
\vskip 1cm{\LARGE\bf
  Connections Between Combinations Without\\
  Specified Separations and Strongly Restricted\\
\vskip .08in 
  Permutations, Compositions, and Bit Strings
}
\vskip 1cm
\large
Michael A. Allen\\
Physics Department\\
Faculty of Science\\
Mahidol University\\
Rama 6 Road\\
Bangkok 10400\\
Thailand\\
\href{mailto:maa5652@gmail.com}{\tt maa5652@gmail.com} \\
\end{center}

\vskip .2 in

\begin{abstract}
Let $S_n$ and $S_{n,k}$ be, respectively, the number of subsets and
$k$-subsets of $\mathbb{N}_n=\{1,\ldots,n\}$ such that no two subset
elements differ by an element of the set $\mathcal{Q}$, the largest
element of which is $q$. We prove a bijection between such $k$-subsets
when $\mathcal{Q}=\{m,2m,\ldots,jm\}$ with $j,m>0$ and permutations
$\pi$ of $\mathbb{N}_{n+jm}$ with $k$ excedances satisfying
$\pi(i)-i\in\{-m,0,jm\}$ for all $i\in\mathbb{N}_{n+jm}$. We also
identify a bijection between another class of restricted permutation
and the cases $\mathcal{Q}=\{1,q\}$ and derive the generating function
for $S_n$ when $q=4,5,6$. We give some classes of
$\mathcal{Q}$ for which $S_n$ is also the number of compositions of
$n+q$ into a given set of allowed parts.
We also prove a bijection between $k$-subsets for a class of
$\mathcal{Q}$ and the set representations of size $k$ of equivalence
classes for the occurrence of a given length-($q+1$) subword within
bit strings. We then formulate a straightforward procedure for
obtaining the generating function for the number of such equivalence
classes.
\end{abstract}

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