论文标题
关于帕斯卡三角概括的两参数家族
On a Two-Parameter Family of Generalizations of Pascal's Triangle
论文作者
论文摘要
我们考虑了一个两参数三角形的家族，其$（n，k）$ - th条目（将初始输入算作$（0,0）$ - th条目）是$ n $板的瓷砖数量（对于任何非简化integer $ n $ niteger $ n $ nime square和$ nime squeares和$ nitege $ n $ n $ n $ nime Square的线性阵列（是$ n $ n $ unit Square）的线性阵列和$（1）$（1） $ m = 1,2，\ dots $和$ t = 2,3，\ dots $，总共使用$ n $ tiles $ k $是梳子。 $（1，M-1; t）$ - 梳子是一个由$ t $单位的正方形子砖（称为牙齿）组成的瓷砖，以便将每个牙齿与下一齿的宽度宽度$ M-1 $组成。我们表明，三角形中的条目是两个连续的广义斐波那契多项式的乘积的系数，该系数每个都提高到某些非负整数功率。我们还提出了$（n+（t-1）m）$ - 带有$ k $ $（1，m-1; t）$ - 梳子的瓷砖之间的两次两者，其余的单元格和$ k $ -subsets的$ \ \ {1，\ ldots，n \} $ a $ $ $ $ $ $ $ $ $ clance n of the $ $ n \} $ a $ $ $ $ c。因此，我们可以提供组合证明此类$ k $ subsets的数量与多项式系数有关。我们还从一类定向的伪扫描上的特定节点中得出了封闭步行数量的递归关系，并应用它获得了有关$ m = 2 $，$ t = 5 $实例的三角形家族的身份。三角形的进一步身份也主要是通过组合证明建立的。
We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are coefficients of the product of two consecutive generalized Fibonacci polynomials each raised to some nonnegative integer power. We also present a bijection between the tiling of an $(n+(t-1)m)$-board with $k$ $(1,m-1;t)$-combs with the remaining cells filled with squares and the $k$-subsets of $\{1,\ldots,n\}$ such that no two elements of the subset differ by a multiple of $m$ up to $(t-1)m$. We can therefore give a combinatorial proof of how the number of such $k$-subsets is related to the coefficient of a polynomial. We also derive a recursion relation for the number of closed walks from a particular node on a class of directed pseudographs and apply it obtain an identity concerning the $m=2$, $t=5$ instance of the family of triangles. Further identities of the triangles are also established mostly via combinatorial proof.
