长度为$n$、带有$x$零和最长$k$的二进制字符串-零的游程
Binary strings of length $n$ with $x$ zeros and longest $k$-runs of zeros
摘要：
本文研究了长度为$n$的二进制字符串的个数$F_{n}(x,k)$，其中包含$x$零和$k$零的最长子字.推导了$F_{n}(x,k)$的递推关系式。我们表示一些已知的数字，如斐波那契，三角形，一些二进制字符串的长度$n$没有$r$ -运行的1和一些组成的$n+1$与最大的和$k+1$方面的$F_{n}(x,k).$类似的结果和应用得到F $_{n}(x,k),$的所有回文二进制字符串的长度$n$包含$x$零和最长的$k$ -运行的零。
英文摘要：
In this paper, we study $F_{n}(x,k)$, the number of binary strings of length $n$ containing $x$ zeros and a longest subword of $k$ zeros. A recurrence relation for $F_{n}(x,k)$ is derived. We expressed few known numbers like Fibonacci, triangular, number of binary strings of length $n$ without $r$-runs of ones and number of compositions of $n+1$ with largest summand $k+1$ in terms of $F_{n}(x,k).$ Similar results and applications were obtained for F$_{n}(x,k),$ the number of all palindromic binary strings of length $n$ containing $x$ zeros and longest $k$-runs of zeros.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
