论文标题
关于帕斯卡三角的两个概括的家族
On Two Families of Generalizations of Pascal's Triangle
论文作者
论文摘要
我们考虑了两个类似帕斯卡的三角形的家族，它们在左侧，右侧为$ M-1 $零的家族。 $ m = 1 $的情况是帕斯卡尔的三角形，当$ m = 2 $时，两个家庭也重合。第一个家庭的成员服从三角形内部的帕斯卡（Pascal）复发。我们表明，也可以通过在$（1/（1-x^m），x/（1-x））$ riordan阵列的每一行中反转元素，并包括$ m $ th的三角形。结果，该家族的特性可以很快获得。 $（n，k）$ - $ m $ $ $的第三个成员是第二个三角家庭的成员，是使用$ k $ $ $ $（1，m-1）$ - 围栏和$ n-k $单位平方的$（n+k）\ times1 $板的瓷砖数量。 $（1，g）$ - 围栏由两个单位正方形子砖组成，该尺寸为宽度$ g $。我们表明，这些三角形的抗异步分子的条目是连续两个斐波那契的能量的系数，并提供了三个系数的$ k $ -subsets $ \ \ \ \ f {1,2，\ ldots，n-m \ \ f sip $ sip a的$ k $ subsets的数量。第二个三角族的其他特性也是通过组合方法获得的。最后，我们为任何像Pascal一样的三角形（或其行转换版本）提供了必要的条件，这些条件是从瓷砖$（n \ times1）$ - 板衍生而来的。
We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the $m$-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$ Riordan array. Properties of this family of triangles can be obtained quickly as a result. The $(n,k)$-th entry in the $m$-th member of the second family of triangles is the number of tilings of an $(n+k)\times1$ board that use $k$ $(1,m-1)$-fences and $n-k$ unit squares. A $(1,g)$-fence is composed of two unit square sub-tiles separated by a gap of width $g$. We show that the entries in the antidiagonals of these triangles are coefficients of products of powers of two consecutive Fibonacci polynomials and give a bijective proof that these coefficients give the number of $k$-subsets of $\{1,2,\ldots,n-m\}$ such that no two elements of a subset differ by $m$. Other properties of the second family of triangles are also obtained via a combinatorial approach. Finally, we give necessary and sufficient conditions for any Pascal-like triangle (or its row-reversed version) derived from tiling $(n\times1)$-boards to be a Riordan array.
