论文标题
没有指定分离的组合
Combinations without specified separations
论文作者
论文摘要
我们考虑$ \ mathbb {n} _n = \ {1,2，\ ldots，n \} $的限制子集，其中$ q \ geq1 $是$ \ mathcal {q} $的最大成员。我们在涉及缺乏指定分离的这种组合的各种问题上获得了新的结果。特别是，当$ | | \ Mathbb {n} _q- \ \ m rathcal {q} | \ leq2 $时，我们发现任何$ \ mathcal {q} $的$ k $ -subsets数量的递归关系。由于我们在此类子集之间给出的两次循环培训，并以$（N+Q）$ - 板（$ n+q）$ - $ n+q $平方单元的线性阵列（单位宽度的线性阵列）（$ 1 \ times1 $ times1 $ tiles）和combs获得的结果（$ n+q $平方单元格的线性阵列（$ n+q $平方单元格），因此以快速而直观的方式获得了结果。 a $（w_1，g_1，w_2，g_2，\ ldots，g_ {t-1}，w_t）$ - 梳子由称为牙齿的$ t $ subtiles组成。梳子中的$ i $ th牙具有宽度$ w_i $，并且与$（i+1）$ - the的牙齿通过宽度$ g_i $分开。在这里，我们仅考虑使用$ w_i，g_i \ in \ mathbb {z}^+$的梳子。当执行具有这样的梳子和正方形的板的限制性重叠瓷砖时，必须将瓷砖的最左键放置在空单元中，而瓷砖中的其余细胞则可以重叠其他已在板上已经已经存在的非最左侧的瓷砖。
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.
