Connections Between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings

Let $S_n$ and $S_{n,k}$ be, respectively, the number of subsets and $k$-subsets of ${\mathbb{N}}_n=\{1,\dots ,n\}$ 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 $\mathcal{Q}=\{m,2m,\dots ,jm\}$ 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.