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

TL;DR

This paper explores combinatorial structures linking restricted subsets, permutations, compositions, and bit strings, establishing bijections and generating functions for various classes of restrictions and configurations.

Contribution

It introduces new bijections between restricted subsets, permutations, and bit string classes, and derives generating functions for these combinatorial objects.

Findings

Bijection between certain restricted subsets and permutations with specific excedance properties.

Explicit generating functions for subset counts when q=4,5,6.

Connections between subset counts and compositions with allowed parts.

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.