论文标题
广义堤防路径中的下步统计
Down-step statistics in generalized Dyck paths
论文作者
论文摘要
在$ k_t $ -DOCK路径中的成对上步之间的下步数，由步骤$ \ {（1，k），（1，-1）\}组成的染色路径的概括，以使该路径在$ y = -t $的线上保持（弱）。结果是通过生成函数的肉豆素证明的，并导致几个有趣的身份以及与其他组合结构的链接。特别是，用于编码理论中使用的卷积卷积代码（二进制矩阵）的$ K_T $ -DYCK路径与穿孔模式之间存在联系。令人惊讶的是，在限制了通常的戴克路径后，这会产生加泰罗尼亚数字的新组合解释。
The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.
