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Combinations without specified separations | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 22، دوره 11، شماره 2، شهریور 2026، صفحه 695-711 اصل مقاله (466.12 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29370.1959 | ||
| نویسنده | ||
| Michael A Allen* | ||
| Physics Department, Faculty of Science, Mahidol University, Rama 6 Road, Bangkok 10400, Thailand | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| restricted combination؛ combinatorial proof؛ tiling؛ directed pseudograph | ||
| مراجع | ||
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