Combinations without specified separations

TL;DR

This paper studies subsets of natural numbers with restricted differences, establishing recursion relations and bijections with restricted-overlap tilings of boards using squares and comb-shaped tiles, to analyze combinatorial properties.

Contribution

It introduces a novel bijection between restricted subsets and overlap tilings with combs, providing new recursion relations for counting such subsets.

Findings

Derived recursion relations for the number of subsets with restricted differences.

Established a bijection between subsets and restricted-overlap tilings with combs.

Provided an intuitive method for analyzing combinatorial structures using tilings.

Abstract

We consider the restricted subsets of $\mathbb{N}_n=\{1,2,\ldots,n\}$ with $q\geq1$ being the largest member of the set $\mathcal{Q}$ of disallowed differences between subset elements. We obtain new results on various classes of problem involving such combinations lacking specified separations. In particular, we find recursion relations for the number of $k$-subsets for any $\mathcal{Q}$ when $|\mathbb{N}_q-\mathcal{Q}|\leq2$. The results are obtained, in a quick and intuitive manner, as a consequence of a bijection we give between such subsets and the restricted-overlap tilings of an $(n+q)$-board (a linear array of $n+q$ square cells of unit width) with squares ($1\times1$ tiles) and combs. A $(w_1,g_1,w_2,g_2,\ldots,g_{t-1},w_t)$-comb is composed of $t$ sub-tiles known as teeth. The $i$-th tooth in the comb has width $w_i$ and is separated from the $(i+1)$-th tooth by a gap of width $g_i$. Here we only consider combs with $w_i,g_i\in\mathbb{Z}^+$. When performing a restricted-overlap tiling of a board with such combs and squares, the leftmost cell of a tile must be placed in an empty cell whereas the remaining cells in the tile are permitted to overlap other non-leftmost filled cells of tiles already on the board.